We show that sharp thresholds for Boolean functions directly imply average-case circuit lower bounds. More formally we show that any Boolean function exhibiting a sharp enough threshold at \emph{arbitrary} critical density cannot be computed by Boolean circuits of bounded depth and polynomial size. Our general result implies new average-case bounded depth circuit lower bounds in a variety of settings. (a) ($k$-cliques) For $k=\Theta(n)$, we prove that any circuit of depth $d$ deciding the presence of a size $k$ clique in a random graph requires exponential-in-$n^{\Theta(1/d)}$ size. To the best of our knowledge, this is the first average-case exponential size lower bound for bounded depth (not necessarily monotone) circuits solving the fundamental $k$-clique problem (for any $k=k_n$). (b)(random 2-SAT) We prove that any circuit of depth $d$ deciding the satisfiability of a random 2-SAT formula requires exponential-in-$n^{\Theta(1/d)}$ size. To the best of our knowledge, this is the first bounded depth circuit lower bound for random $k$-SAT for any value of $k \geq 2.$ Our results also provide the first rigorous lower bound in agreement with a conjectured, but debated, ``computational hardness'' of random $k$-SAT around its satisfiability threshold. (c)(Statistical estimation -- planted $k$-clique) Over the recent years, multiple statistical estimation problems have also been proven to exhibit a ``statistical'' sharp threshold, called the All-or-Nothing (AoN) phenomenon. We show that AoN also implies circuit lower bounds for statistical problems. As a simple corollary of that, we prove that any circuit of depth $d$ that solves to information-theoretic optimality a ``dense'' variant of the celebrated planted $k$-clique problem requires exponential-in-$n^{\Theta(1/d)}$ size.

翻译：我们证明，布尔函数的尖锐阈值直接蕴含平均情况下的电路下界。更精确地说，我们证明任何在任意临界密度下展现出足够尖锐阈值的布尔函数，都无法被有界深度且多项式规模的布尔电路计算。该通用结果在多种场景下推导出新的平均情况有界深度电路下界：(a) (k-团问题) 对于k=Θ(n)，我们证明判断随机图中是否存在规模为k的团的深度为d的电路，其规模需关于n^{Θ(1/d)}呈指数级增长。据我们所知，这是首个针对有界深度（非单调）电路解决基本k-团问题（对任意k=k_n）的平均情况指数级规模下界。(b) (随机2-SAT) 我们证明判断随机2-SAT公式可满足性的深度为d的电路，其规模需关于n^{Θ(1/d)}呈指数级增长。据我们所知，这是首个对任意k≥2的随机k-SAT问题建立的有界深度电路下界。该结果还首次严格验证了随机k-SAT在其可满足性阈值附近存在（曾被提出但存在争议的）“计算困难性”猜想。(c) (统计估计——植入k-团问题) 近年来，多种统计估计问题被证明存在名为“全或无”(AoN)的统计尖锐阈值。我们证明AoN现象同样蕴含统计问题的电路下界。作为直接推论，我们证明：任何深度为d的电路，若要以信息论最优方式解决著名植入k-团问题的“密集”变体，其规模需关于n^{Θ(1/d)}呈指数级增长。