Extended low-degree conjecture for general hypothesis testing problems

Prove the extended low-degree conjecture for general hypothesis testing: For distributions P and Q over inputs of size n and any δ ∈ [0,1], show that every [0,1]-valued function f computable in time exp(n^δ) satisfies R_{P,Q}(f) ≲ max over all polynomials g of degree at most n^δ of R_{P,Q}(g), where R_{P,Q}(h) = (E_{Y∼P}[h(Y)] − E_{Y∼Q}[h(Y)]) / sqrt(Var_{Y∼Q}(h(Y))).

Background

To extend their lower bounds beyond constant numbers of communities, the authors invoke a stronger version of the low-degree conjecture that connects the distinguishing power of any subexponential-time algorithm to that of low-degree polynomials for arbitrary P vs Q testing problems.

This extended conjecture strengthens the original by asserting a tight comparison inequality between the best exp(n^δ)-time test and the best degree n^δ polynomial test, and is related to low-degree bounds for random optimization problems. It is a central unproven assumption for the paper’s results when the number of blocks k can grow with n.