Informal low-degree likelihood-ratio conjecture

Let $\mathbb{P}_n$ and $\mathbb{Q}_n$ be probability distributions on $\Omega_n$, with likelihood ratio $L_n=d\mathbb{P}_n/d\mathbb{Q}_n$. For a degree bound $D$, let $L_n^{\le D}$ denote the orthogonal projection of $L_n$ onto the degree-$D$ polynomials under the $L^2(\mathbb{Q}_n)$ inner product.

Informal low-degree conjecture. For “nice” distributions $\mathbb{P}_n$ and $\mathbb{Q}_n$, if

$$\|L_n^{\le D}\|_{L^2(\mathbb{Q}_n)}=O(1)$$

for some $D=\log^{1+\Omega(1)}(n)$, then there is no randomized polynomial-time algorithm for strong detection between $\mathbb{P}$ and $\mathbb{Q}$.

The conjecture formalizes the heuristic that bounded low-degree likelihood-ratio norm rules out efficient strong detection. It is explicitly informal because the paper does not define “nice” distributions; precise variants use coordinate degree. Evidence and applications are discussed for models including the stochastic block model and spiked tensor model, while the converse is not expected to hold.