Low-degree conjecture for computational indistinguishability

Let $\mathbb{P}_{{\cal H}_0}$ and $\mathbb{P}_{{\cal H}_1}$ be the null and planted distributions, let ${\cal L}_n$ be their likelihood ratio, and let ${\cal L}_{n,\le D}$ be its orthogonal projection onto the space of polynomial functions of total degree at most $D$ with respect to $\mathbb{P}_{{\cal H}_0}$. Strong detection means distinguishing $\mathbb{P}_{{\cal H}_0}$ from $\mathbb{P}_{{\cal H}_1}$.

Low-degree conjecture. If there exist $\epsilon>0$ and $D=D(n)\geq (\log n)^{1+\epsilon}$ such that

$$\|{\cal L}_{n,\leq D}\|_{{\cal H}_0}$$

remains bounded as $n\to\infty$, then no polynomial-time algorithm can distinguish $\mathbb{P}_{{\cal H}_0}$ from $\mathbb{P}_{{\cal H}_1}$, that is, achieve strong detection.

The conjecture proposes that bounded low-degree likelihood-ratio norm characterizes computational hardness once the degree grows slightly faster than logarithmic. It is presented as an informal principle, and the supplied source gives no resolution.