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1. Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols

   https://doi.org/10.4230/LIPIcs.CCC.2023.17  

   van Melkebeek, Dieter; Mocelin Sdroievski, Nicollas (July 2023, Proceedings of the 38th Computational Complexity Conference (CCC 2023))

   Ta-Shma, Amnon (Ed.)

   A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS'21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n^a. We show that if every Arthur-Merlin protocol that runs in time n^c for c = O(log² a) can only compute f correctly on finitely many inputs, then AM is in NP. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM, as well as an unconditional mild derandomization result for AM. 

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2. Instance-Wise Hardness and Refutation versus Derandomization for Arthur-Merlin Protocols

   https://doi.org/10.1007/s00037-025-00279-2  

   van_Melkebeek, Dieter; Mocelin_Sdroievski, Nicollas (December 2025, computational complexity)

   Abstract A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settingsblackboxderandomization, i.e., derandomization through pseudorandom generators, has been shown equivalent to lower bounds for decision problems against circuits.Recently, Chen and Tell (FOCS'21) established nearequivalences in the BPP setting betweenwhiteboxderandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function$$f$$ $f$on the given instance. Follow-up works managed to obtain full equivalences in the BPP setting by exploiting acompressionproperty of classical pseudorandom generator constructions. In particular, Chen, Tell, and Williams (FOCS'23) showed that derandomization of BPP is equivalent toconstructivelower bounds against algorithms that go through a compression phase.In this paper, we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness-to-derandomization direction, consider a length-preserving function$$f$$ $f$computable by a nondeterministic algorithm that runs in time$$n^a$$ $n^a$. We show that if every Arthur-Merlin protocol that runs in time$$n^c$$ $n^c$for$$c=O(\log^2 a)$$ $c=O\left({log}^{2}a\right)$can only compute$$f$$ $f$correctly on finitely many inputs, then AM is in NP. We also obtain equivalences between constructive lower bounds against Arthur-Merlin protocols that go through a compression phase and derandomization of AM viatargetedgenerators. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs of proximity for nondeterministic computations. As a by-product of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). By-products in the average-case setting include the first uniform hardness vs. randomness trade-offs for AM, as well as an unconditional mild derandomization result for AM. 

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3. New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification

   https://doi.org/10.4230/LIPIcs.ITCS.2024.53  

   Ghosh, Prantar; Shah, Vihan (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Guruswami, Venkatesan (Ed.)

   We present novel lower bounds in the Merlin-Arthur (MA) communication model and the related annotated streaming or stream verification model. The MA communication model extends the classical communication model by introducing an all-powerful but untrusted player, Merlin, who knows the inputs of the usual players, Alice and Bob, and attempts to convince them about the output. We focus on the online MA (OMA) model where Alice and Merlin each send a single message to Bob, who needs to catch Merlin if he is dishonest and announce the correct output otherwise. Most known functions have OMA protocols with total communication significantly smaller than what would be needed without Merlin. In this work, we introduce the notion of non-trivial-OMA complexity of a function. This is the minimum total communication required when we restrict ourselves to only non-trivial protocols where Alice sends Bob fewer bits than what she would have sent without Merlin. We exhibit the first explicit functions that have this complexity superlinear - even exponential - in their classical one-way complexity: this means the trivial protocol, where Merlin communicates nothing and Alice and Bob compute the function on their own, is exponentially better than any non-trivial protocol in terms of total communication. These OMA lower bounds also translate to the annotated streaming model, the MA analogue of single-pass data streaming. We show large separations between the classical streaming complexity and the non-trivial annotated streaming complexity (for the analogous notion in this setting) of fundamental problems such as counting distinct items, as well as of graph problems such as connectivity and k-connectivity in a certain edge update model called the support graph turnstile model that we introduce here. 

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4. Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

   https://doi.org/10.1007/s00453-023-01102-6  

   Akmal, Shyan; Chen, Lijie; Jin, Ce; Raj, Malvika; Williams, Ryan (February 2023, Algorithmica)

   Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$ $\tilde{O}\left(n\right)$. Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$time).Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $\tilde{O}\left(n^{\lceil k/2\rceil }\right)$(where$$k\ge 3$$ $k\ge 3$). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$ $\tilde{O}\left(m\right)$. Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$ $\tilde{O}\left(n^2\right)$. Note this is optimal, as the matrix can have$$\Omega (n^2)$$ $\Omega \left(n^2\right)$nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$ $\tilde{O}\left(n^{2.94}\right)$nondeterministic time algorithm.Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$ $2^{n/2-n/O\left(k\right)}$. We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$ $2^{n/2}·\text{poly}\left(n\right)$-time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$ $\#$SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$ $2^{n/2}/n^{\omega \left(1\right)}$time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$ $2^{4n/5}·\text{poly}\left(n\right)$. Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$ $2^{2n/3}·\text{poly}\left(n\right)$time.Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$ $2^{n/3}·\text{poly}\left(n\right)$, improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$ $2^{0.49991n}·\text{poly}\left(n\right)$time. 

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5. Opening Up the Distinguisher: A Hardness to Randomness Approach for BPL = L that Uses Properties of BPL

   Doron, Dean; Pyne, Ted; Tell, Roei (June 2024, 56th Annual ACM Symposium on Theory of Computing (STOC 2024))

   We provide compelling evidence for the potential of hardness-vs.-randomness approaches to make progress on the long-standing problem of derandomizing space-bounded computation. Our first contribution is a derandomization of bounded-space machines from hardness assumptions for classes of uniform deterministic algorithms, for which strong (but non-matching) lower bounds can be unconditionally proved. We prove one such result for showing that BPL=L “on average”, and another similar result for showing that BPSPACE[O(n)]=DSPACE[O(n)]. Next, we significantly improve the main results of prior works on hardness-vs.-randomness for logspace. As one of our results, we relax the assumptions needed for derandomization with minimal memory footprint (i.e., showing BPSPACE[S]⊆ DSPACE[c · S] for a small constant c), by completely eliminating a cryptographic assumption that was needed in prior work. A key contribution underlying all of our results is non-black-box use of the descriptions of space-bounded Turing machines, when proving hardness-to-randomness results. That is, the crucial point allowing us to prove our results is that we use properties that are specific to space-bounded machines. 

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