Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural "Sure/Shor separator"---that is, a set of quantum states that can account for all experiments performed to date, but not for Shor's factoring algorithm. We propose as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require n## additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future.

For any AND-OR formula of size N, there exists a bounded-error N 1 2 +o(1)-time quantum algorithm, based on a discrete-time quantum walk, that evaluates this formula on a black-box input. Balanced, or “approximately balanced,” formulas can be evaluated in O ( √ N) queries, which is optimal. It follows that the (2 − o(1))th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy. 1

We propose Gate Evaluation Secret Sharing (GESS) -- a new kind of secret sharing, designed for use in secure function evaluation (SFE) with minimal interaction. The resulting simple and powerful GESS approach to SFE is a generalization of Yao's garbled circuit technique.

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Abstract. Spira [28] showed that any Boolean formula of size s can be simulated in depth O(log s). We generalize Spira’s theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s) log s). If the segregator size is at least s ε for some constant ε> 0, then we can obtain a simulation of depth O(f(s)). This improves and generalizes a simulation of polynomial-size Boolean circuits of constant treewidth k in depth O(k 2 log n) by Jansen and Sarma [17]. Since the existence of small balanced separators in a directed acyclic graph implies that the graph also has small segregators, our results also apply to circuits with small separators. Our results imply that the class of languages computed by non-uniform families of polynomial-size circuits that have constant size segregators equals non-uniform NC 1. Considering space bounded Turing machines to generate the circuits, for f(s) log 2 s-space uniform families of Boolean circuits our small-depth simulations are also f(s) log 2 s-space uniform. As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators (or separators) is in deterministic SP ACE(log 2 n). Our results also imply that the Planar Circuit Value problem, which is known to be P-Complete [16], can be solved in deterministic SP ACE ( √ n log n). Key words: Boolean circuits, circuit size, circuit depth, Spira’s theorem, Turing machines, space complexity 1