The Polynomial-Time Low-Degree Conjecture Is False

Researchers have disproved the polynomial-time low-degree conjecture, a widely used assumption in average-case inference and high-dimensional statistics, by constructing a family of permutation-invariant distributions on simple graphs that are indistinguishable from uniform by low-degree tests yet can be distinguished in polynomial time. For every fixed integer r≥3, the distribution matches G(n,1/2) on all subgraphs of size D_n=Θ((log n)^{r-1}) edges, but after independent resampling at a fixed positive rate, a deterministic rank test succeeds in polynomial time. The result shows that low-degree indistinguishability, uniform null distribution, permutation invariance, and independent resampling do not imply polynomial-time hardness.

Computer Science Computational Complexity Submitted on 22 Jul 2026 Title:The Polynomial-Time Low-Degree Conjecture is False View PDF /pdf/2607.20318 HTML experimental https://arxiv.org/html/2607.20318v1 Abstract:The low-degree method and its associated lower bounds are widely used to guide algorithm design and to provide evidence of computational hardness in average-case inference, high-dimensional statistics, random optimization, and related problems. This led to the low-degree conjecture, which predicts that when the low-degree advantage between a planted distribution and a uniform null distribution remains bounded, no efficient distinguisher can succeed after independent noise, provided that the planted distribution has permutation symmetry. Several works have produced counterexamples to variants of this conjecture or to versions for algorithms with higher time complexity, but the conjecture remained open in its standard binary, polynomial-time formulation. We disprove the polynomial-time low-degree conjecture by giving a family of examples in this setting. For every fixed integer $r\geq3$, we construct a permutation-invariant distribution $\mathbb{P} n$ on simple graphs, with $\mathbb{Q} n=G n,1/2 $, such that every marginal of $\mathbb{P} n$ on at most $D n=\Theta \log n ^{r-1} $ edges is uniform. Therefore, the low-degree advantage is zero through degree $D n$. Nevertheless, after every edge is independently resampled at a fixed positive rate, a deterministic rank test strongly distinguishes the resulting distribution from $\mathbb{Q} n$ in polynomial time. The construction chooses a subspace of a Reed--Muller code whose nonzero polynomials have small absolute bias, selects points whose evaluation vectors have no short linear dependencies, and evaluates a random alternating bilinear form on pairs of these vectors. Our result shows that low-degree indistinguishability, a uniform null distribution, permutation invariance, and independent resampling do not by themselves imply polynomial-time hardness, and suggests that a valid general conjecture must impose an additional condition. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .