Proving complexity lower bounds remains a challenging task: currently, we only know how to prove conditional uniform (algorithm) lower bounds and nonuniform (circuit) lower bounds in restricted circuit models. About a decade ago, Williams (STOC 2010) showed how to derive nonuniform lower bounds from uniform \emph{upper} bounds. Since then, a number of results of this kind have been proved. For example, Jahanjou et al. (ICALP 2015) and Carmosino et al. (ITCS 2016) proved that if NSETH fails, then $\text{E}^{\text{NP}}$ has circuit size $\omega(n)$. Just recently, it was shown how uniform \emph{lower} bounds can be used to derive nonuniform lower bounds: Belova et al. (SODA 2024) showed that if MAX-3-SAT cannot be solved in co-nondeterministic time $O(2^{(1 - \varepsilon)n})$, then there exists an explicit polynomial family of high arithmetic circuit size; Williams (FOCS 2024) showed that if NSETH is true, then Boolean Inner Product requires large $\text{ETHR} \circ \text{ETHR}$ circuits. In this paper, we continue developing this line of research and show that nondeterministic uniform lower bounds imply nonuniform lower bounds for various types of objects that are notoriously hard to analyze: circuits, matrix rigidity, and tensor rank. Specifically, we prove that if NSETH is true, then there exists a monotone Boolean function family in coNP of monotone circuit size $2^{\Omega(n / \log n)}$. Combining this with the result above, we get win-win circuit lower bounds: either $\text{E}^{\text{NP}}$ requires circuits of size $w(n)$ or coNP requires monotone circuits of size $2^{\Omega(n / \log n)}$. We also prove that MAX-3-SAT cannot be solved in co-nondeterministic time $O(2^{(1 - \varepsilon)n})$, then there exist small explicit families of high rigidity matrices and high rank tensors.

翻译：证明复杂性下界仍然是一项具有挑战性的任务：目前，我们仅知道如何在受限电路模型中证明条件性均匀（算法）下界和非均匀（电路）下界。大约十年前，Williams (STOC 2010) 展示了如何从均匀上界推导非均匀下界。此后，此类结果被多次证明。例如，Jahanjou 等人 (ICALP 2015) 和 Carmosino 等人 (ITCS 2016) 证明了如果 NSETH 不成立，则 $\text{E}^{\text{NP}}$ 具有电路规模 $\omega(n)$。就在最近，研究展示了如何利用均匀下界推导非均匀下界：Belova 等人 (SODA 2024) 证明了如果 MAX-3-SAT 无法在共非确定性时间 $O(2^{(1 - \varepsilon)n})$ 内求解，则存在一个具有高算术电路规模的显式多项式族；Williams (FOCS 2024) 证明了如果 NSETH 成立，则布尔内积需要大规模的 $\text{ETHR} \circ \text{ETHR}$ 电路。在本文中，我们继续推进这一研究方向，并证明非确定性均匀下界意味着多种难以分析对象（电路、矩阵刚性和张量秩）的非均匀下界。具体而言，我们证明如果 NSETH 成立，则存在一个属于 coNP 的单调布尔函数族，其单调电路规模为 $2^{\Omega(n / \log n)}$。结合上述结果，我们得到了双赢的电路下界：要么 $\text{E}^{\text{NP}}$ 需要规模为 $w(n)$ 的电路，要么 coNP 需要规模为 $2^{\Omega(n / \log n)}$ 的单调电路。我们还证明，如果 MAX-3-SAT 无法在共非确定性时间 $O(2^{(1 - \varepsilon)n})$ 内求解，则存在小规模的显式高刚性矩阵族和高秩张量族。