The low-degree polynomial conjecture on computational indistinguishability

Let $\mathbb{P}_n$ and $\mathbb{Q}_n$ denote a sequence of probability measures with sample space $\mathbb{R}^{n^k}$, where $k=O(1)$. Suppose that every polynomial $f$ of degree $O(\log n)$ satisfying

$$\mathbb{E}_{\mathbb{Q}_n} f^2=1$$

is bounded under $\mathbb{P}_n$ with high probability as $n\to\infty$, and that some further regularity conditions hold. Low-degree polynomial conjecture. There is no polynomial-time test distinguishing $\mathbb{P}_n$ from $\mathbb{Q}_n$ with both type I and type II errors tending to $0$ as $n\to\infty$. The conjecture is a proposed computational lower-bound principle: control of all low-degree polynomials should rule out efficient tests, subject to the stated regularity conditions.