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We prove that for some constant a > 1, for all k ≤ a, MATIME[n^{k(1+o(1))}]/1 ⊄ SIZE[O(n^k)], for some specific o(1) function. This is a super linear polynomial circuit lower bound. Previously, Santhanam [Santhanam, 2007] showed that there exists a constant c>1 such that for all k>1: MATIME[n^{ck}]/1 ⊄ SIZE[O(n^k)]. Inherently to Santhanam’s proof, c is a large constant and there is no upper bound on c. Using ideas from Murray and Williams [Murray and Williams, 2018], one can show for all k>1: MATIME [n^{10 k²}]/1 ⊄ SIZE[O(n^k)]. To prove this result, we construct the first PCP for SPACE[n] with quasi-linear verifier time: our PCP has a Õ(n) time verifier, Õ(n) space prover, O(log(n)) queries, and polynomial alphabet size. Prior to this work, PCPs for SPACE[O(n)] had verifiers that run in Ω(n²) time. This PCP also proves that NE has MIP verifiers which run in time Õ(n). 

We give the first explicit constant rate, constant relative distance, linear codes with an encoder that runs in time n^{1 + o(1)} and space polylog(n) provided random access to the message. Prior to this work, the only such codes were non-explicit, for instance repeat accumulate codes [Divsalar et al., 1998] and the codes described in [Gál et al., 2013]. To construct our codes, we also give explicit, efficiently invertible, lossless condensers with constant entropy gap and polylogarithmic seed length. In contrast to encoders with random access to the message, we show that encoders with sequential access to the message can not run in almost linear time and polylogarithmic space. Our notion of sequential access is much stronger than streaming access. 

In a regular PCP the verifier queries each proof symbol in the same number of tests. This number is called the degree of the proof, and it is at least 1/(sq) where s is the soundness error and q is the number of queries. It is incredibly useful to have regularity and reduced degree in PCP. There is an expander-based transformation by Papadimitriou and Yannakakis that transforms any PCP with a constant number of queries and constant soundness error to a regular PCP with constant degree. There are also transformations for low error projection and unique PCPs. Other PCPs are constructed especially to be regular. In this work we show how to regularize and reduce degree of PCPs with a possibly large number of queries and low soundness error. As an application, we prove NP-hardness of an unweighted variant of the collective minimum monotone satisfying assignment problem, which was introduced by Hirahara (FOCS'22) to prove NP-hardness of MCSP^* (the partial function variant of the Minimum Circuit Size Problem) under randomized reductions. We present a simplified proof and sufficient conditions under which MCSP^* is NP-hard under the standard notion of reduction: MCSP^* is NP-hard under deterministic polynomial-time many-one reductions if there exists a function in E that satisfies certain direct sum properties. 

The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time Õ(n+S²). This improves on the best previous bound of Õ(n+S³) and matches the result for deterministic space bounded algorithms, up to polylog(S) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T, space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time Õ(n + S log(T) + S d) and the prover runs in time 2^O(S). For d = O(log(T)), this matches the best known interactive proofs for deterministic algorithms, up to polylog(S) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log(T). We also improve the best prior verifier time for unbounded alternations by a factor of S. Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in. 

A Chor–Goldreich (CG) source is a sequence of random variables X = X1 ∘ … ∘ Xt, where each Xi ∼ {0,1}d and Xi has δ d min-entropy conditioned on any fixing of X1 ∘ … ∘ Xi−1. The parameter 0<δ≤ 1 is the entropy rate of the source. We typically think of d as constant and t as growing. We extend this notion in several ways, defining almost CG sources. Most notably, we allow each Xi to only have conditional Shannon entropy δ d. We achieve pseudorandomness results for almost CG sources which were not known to hold even for standard CG sources, and even for the weaker model of Santha–Vazirani sources: We construct a deterministic condenser that on input X, outputs a distribution which is close to having constant entropy gap, namely a distribution Z ∼ {0,1}m for m ≈ δ dt with min-entropy m−O(1). Therefore, we can simulate any randomized algorithm with small failure probability using almost CG sources with no multiplicative slowdown. This result extends to randomized protocols as well, and any setting in which we cannot simply cycle over all seeds, and a “one-shot” simulation is needed. Moreover, our construction works in an online manner, since it is based on random walks on expanders. Our main technical contribution is a novel analysis of random walks, which should be of independent interest. We analyze walks with adversarially correlated steps, each step being entropy-deficient, on good enough lossless expanders. We prove that such walks (or certain interleaved walks on two expanders), starting from a fixed vertex and walking according to X1∘ … ∘ Xt, accumulate most of the entropy in X.