The low-degree conjecture for strong detection

Let $\mathcal{Q}$ and $\mathcal{P}$ be sufficiently nice distributions, and let $\operatorname{Adv}_{\leq D}(\mathcal{Q},\mathcal{P})$ denote the degree-$D$ low-degree advantage for distinguishing them. Suppose that there exists $\varepsilon>0$ and $D=D(n)$ satisfying

$$D\geq (\log n)^{1+\varepsilon}$$

and that $\operatorname{Adv}_{\leq D}(\mathcal{Q},\mathcal{P})$ remains bounded as $n\to\infty$. Low-degree conjecture. There is no polynomial-time algorithm that achieves strong detection between $\mathcal{Q}$ and $\mathcal{P}$. This is an informal version of the low-degree conjecture attributed to Hopkins. It proposes that bounded low-degree advantage at sufficiently high degree rules out efficient strong detection, under the unspecified regularity condition that the distributions are sufficiently nice.