The Quasi-Polynomial Low-Degree Conjecture is False
Rares-Darius Buhai, Jun-Ting Hsieh, Aayush Jain, Pravesh K. Kothari
TL;DR
This work disproves the Low-Degree Conjecture by constructing counterexamples where the degree-$D$ low-degree advantage vanishes (for $D = n^{1-O( ext{ε})}$ with fixed $ ext{ε}>0$) yet a noise-tolerant distinguisher exists in quasi-polynomial time. It presents two complementary counterexamples: (i) a Boolean, permutation-invariant planted-null pair built via permutation-resilient, list-decodable codes (rooted in noisy polynomial interpolation) that yields a polytime distinguisher in the rectangular setting, and (ii) a rotationally invariant matrix model where the top eigenvalue serves as a distinguishing statistic despite vanishing LDA for large $D$. The results draw on Reed-Solomon list-decoding and eigenvalue concentration tools (Hanson–Wright) and show that vanishing LDA does not universally imply hardness, motivating a more nuanced theory of average-case complexity and urging refined conjectures or domain-specific analyses. Overall, the paper exposes limitations of the low-degree heuristic and guides future work toward conditional hardness results that account for symmetry, noise models, and problem structure.
Abstract
There is a growing body of work on proving hardness results for average-case estimation problems by bounding the low-degree advantage (LDA) - a quantitative estimate of the closeness of low-degree moments - between a null distribution and a related planted distribution. Such hardness results are now ubiquitous not only for foundational average-case problems but also central questions in statistics and cryptography. This line of work is supported by the low-degree conjecture of Hopkins, which postulates that a vanishing degree-$D$ LDA implies the absence of any noise-tolerant distinguishing algorithm with runtime $n^{\widetilde{O}(D)}$ whenever 1) the null distribution is product on $\{0,1\}^{\binom{n}{k}}$, and 2) the planted distribution is permutation invariant, that is, invariant under any relabeling $[n] \rightarrow [n]$. In this paper, we disprove this conjecture. Specifically, we show that for any fixed $\varepsilon>0$ and $k\geq 2$, there is a permutation-invariant planted distribution on $\{0,1\}^{\binom{n}{k}}$ that has a vanishing degree-$n^{1-O(\varepsilon)}$ LDA with respect to the uniform distribution on $\{0,1\}^{\binom{n}{k}}$, yet the corresponding $\varepsilon$-noisy distinguishing problem can be solved in $n^{O(\log^{1/(k-1)}(n))}$ time. Our construction relies on algorithms for list-decoding for noisy polynomial interpolation in the high-error regime. We also give another construction of a pair of planted and (non-product) null distributions on $\mathbb{R}^{n \times n}$ with a vanishing $n^{Ω(1)}$-degree LDA while the largest eigenvalue serves as an efficient noise-tolerant distinguisher. Our results suggest that while a vanishing LDA may still be interpreted as evidence of hardness, developing a theory of average-case complexity based on such heuristics requires a more careful approach.