The low-degree polynomial test conjecture

Consider a broad class of hypothesis-testing problems $H_0$ versus $H_1$. A polynomial $f$ is a $D$-simple statistic if it has degree at most $D$ and satisfies

$$\mathbb{E}_{H_0}f(X)=0,\qquad \mathbb{E}_{H_0}f(X)^2=1,$$

while $\mathbb{E}_{H_1}f(X)\to\infty$. Low-degree polynomial test conjecture. There is a test running in time $n^{\tilde O(D)}$ with Type I plus Type II errors tending to zero if and only if there is a successful $D$-simple statistic. This conjecture proposes that low-degree tests capture the power of the sum-of-squares hierarchy and, more generally, all efficient hypothesis-testing algorithms; it is presented as a conjecture based on heuristic understanding and is unresolved in the stated broad form.