Over the past decade, the low-degree heuristic has been used to estimate the algorithmic thresholds for a wide range of average-case planted vs null distinguishing problems. Such results rely on the hypothesis that if the low-degree moments of the planted and null distributions are sufficiently close, then no efficient (noise-tolerant) algorithm can distinguish between them. This hypothesis is appealing due to the simplicity of calculating the low-degree likelihood ratio (LDLR) -- a quantity that measures the similarity between low-degree moments. However, despite sustained interest in the area, it remains unclear whether low-degree indistinguishability actually rules out any interesting class of algorithms. In this work, we initiate the study and develop technical tools for translating LDLR upper bounds to rigorous lower bounds against concrete algorithms. As a consequence, we prove: for any permutation-invariant distribution $\mathsf{P}$, 1. If $\mathsf{P}$ is over $\{0,1\}^n$ and is low-degree indistinguishable from $U = \mathrm{Unif}(\{0,1\}^n)$, then a noisy version of $\mathsf{P}$ is statistically indistinguishable from $U$. 2. If $\mathsf{P}$ is over $\mathbb{R}^n$ and is low-degree indistinguishable from the standard Gaussian ${N}(0, 1)^n$, then no statistic based on symmetric polynomials of degree at most $O(\log n/\log \log n)$ can distinguish between a noisy version of $\mathsf{P}$ from ${N}(0, 1)^n$. 3. If $\mathsf{P}$ is over $\mathbb{R}^{n\times n}$ and is low-degree indistinguishable from ${N}(0,1)^{n\times n}$, then no constant-sized subgraph statistic can distinguish between a noisy version of $\mathsf{P}$ and ${N}(0, 1)^{n\times n}$.

翻译：在过去十年中，低阶启发式准则被广泛用于估计各类平均情况下的"植入分布与零分布区分问题"的算法阈值。此类结果依赖于以下假设：若植入分布与零分布的低阶矩足够接近，则不存在高效（噪声容忍）算法能够区分二者。这一假设因其计算简便性而颇具吸引力——只需计算低阶似然比（LDLR）这一衡量低阶矩相似度的量即可。然而，尽管该领域持续受到关注，低阶不可区分性是否确实能排除某些有意义的算法类别仍不明确。在本研究中，我们开创性地探索了将LDLR上界转化为针对具体算法的严格下界的技术工具。作为推论，我们证明了：对于任意置换不变的分布$\mathsf{P}$，1. 若$\mathsf{P}$定义在$\{0,1\}^n$上且与均匀分布$U = \mathrm{Unif}(\{0,1\}^n)$低阶不可区分，则$\mathsf{P}$的噪声版本与$U$在统计上不可区分。2. 若$\mathsf{P}$定义在$\mathbb{R}^n$上且与标准高斯分布${N}(0, 1)^n$低阶不可区分，则任何基于次数不超过$O(\log n/\log \log n)$的对称多项式的统计量，均无法区分$\mathsf{P}$的噪声版本与${N}(0, 1)^n$。3. 若$\mathsf{P}$定义在$\mathbb{R}^{n\times n}$上且与${N}(0,1)^{n\times n}$低阶不可区分，则任何常数规模的子图统计量均无法区分$\mathsf{P}$的噪声版本与${N}(0, 1)^{n\times n}$。