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Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Our proof uses elementary counting together with a novel connection to the sunflower lemma. In addition to a simplified proof, our approach opens up a new avenue of attack towards proving lifting theorems with improved gadget size - one of the main challenges in the area. Focusing on one of the most widely used gadgets - the index gadget - existing lifting techniques are known to require at least a quadratic gadget size. Our new approach combined with robust sunflower lemmas allows us to reduce the gadget size to near linear. We conjecture that it can be further improved to polylogarithmic, similar to the known bounds for the corresponding robust sunflower lemmas. 

Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight ℓ2-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX. Towards the latter, we introduce a novel spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank as a parameter controlling the performance of algorithms on structured objects. Combined with a sum-of-squares proof for the former ℓ2-characterization, we give a concrete application of this framework to algorithms for unique games on HD-walks, where in many cases we improve the state of the art (Barak, Raghavendra, and Steurer FOCS 2011, and Arora, Barak, and Steurer JACM 2015) from nearly-exponential to polynomial time (e.g. for sparsifications of Johnson graphs or of slices of the q-ary hypercube). Our characterization of expansion also holds an interesting connection to hardness of approximation, where an ℓ∞-variant for the Grassmann graphs was recently used to resolve the 2-2 Games Conjecture (Khot, Minzer, and Safra FOCS 2018). We give a reduction from a related ℓ∞-variant to our ℓ2-characterization, but it loses factors in the regime of interest for hardness where the gap between ℓ2 and ℓ∞ structure is large. Nevertheless, our results open the door for further work on the use of HDX in hardness of approximation and their general relation to unique games.