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Electronic Colloquium on Computational Complexity

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TR23-178 | 16th November 2023
Louis Golowich, Tali Kaufman

NLTS Hamiltonians and Strongly-Explicit SoS Lower Bounds from Low-Rate Quantum LDPC Codes

Recent constructions of the first asymptotically good quantum LDPC (qLDPC) codes led to two breakthroughs in complexity theory: the NLTS (No Low-Energy Trivial States) theorem (Anshu, Breuckmann, and Nirkhe, STOC'23), and explicit lower bounds against a linear number of levels of the Sum-of-Squares (SoS) hierarchy (Hopkins and Lin, FOCS'22).

In ... more >>>


TR23-177 | 18th November 2023
Kiran Kedlaya, Swastik Kopparty

On the degree of polynomials computing square roots mod p

Revisions: 1

For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$.

When $p \equiv 3$ mod $4$, it is well known that $f(X) = X^{(p+1)/4}$ computes square ... more >>>


TR23-176 | 15th November 2023
William Hoza

A Technique for Hardness Amplification Against $\mathrm{AC}^0$

Revisions: 2

We study hardness amplification in the context of two well-known "moderate" average-case hardness results for $\mathrm{AC}^0$ circuits. First, we investigate the extent to which $\mathrm{AC}^0$ circuits of depth $d$ can approximate $\mathrm{AC}^0$ circuits of some larger depth $d + k$. The case $k = 1$ is resolved by HÃ¥stad, Rossman, ... more >>>



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