We initiate a systematic study of the computational complexity of property testing, focusing on the relationship between query and time complexity. While traditional work in property testing has emphasized query complexity, relatively little is known about the computational hardness of property testers. Our goal is to chart the landscape of time-query interplay and develop tools for proving time complexity lower bounds. Our first contribution is a pair of time-query hierarchy theorems for property testing. For all suitable nondecreasing functions $q(n)$ and $t(n)$ with $t(n)\geq q(n)$, we construct properties with query complexity $\tildeΘ(q(n))$ and time complexity $\tildeΩ(t(n))$. Our weak hierarchy holds unconditionally, whereas the strong version-assuming the Strong Exponential Time Hypothesis-provides better control over the time complexity of the constructed properties. We then turn to halfspaces in $\mathbb{R}^d$, a fundamental class in property testing and learning theory. We study the problem of approximating the distance from the input function to the nearest halfspace within additive error $ε$. For the distribution-free distance approximation problem, known algorithms achieve query complexity $O(d/ε^2)$, but take time $\tildeΘ(1/ε^d)$. We provide a fine-grained justification for this gap: assuming the $k$-SUM conjecture, any algorithm must have running time $Ω(1/ε^{d/2})$. This fine-grained lower bound yields a provable separation between query and time complexity for a natural and well-studied (tolerant) testing problem. We also prove that any Statistical Query (SQ) algorithm under the standard Gaussian distribution requires $(1/ε)^{Ω(d)}$ queries if the queries are answered with additive error up to $ε^{Ω(d)}$, revealing a fundamental barrier even in the distribution-specific setting.

翻译：我们启动了对性质测试中计算复杂性的系统性研究，重点关注查询复杂度与时间复杂度的关系。虽然性质测试的传统工作强调查询复杂度，但关于性质测试器的计算难度知之甚少。我们的目标是描绘时间-查询交互的图景，并开发用于证明时间复杂度下界的工具。我们的第一个贡献是为性质测试提供了一对时间-查询层次定理。对于所有适当的非递减函数 $q(n)$ 和 $t(n)$（满足 $t(n)\geq q(n)$），我们构造了查询复杂度为 $\tildeΘ(q(n))$ 且时间复杂度为 $\tildeΩ(t(n))$ 的性质。弱层次定理无条件成立，而强版本（假设强指数时间假说）则对所构造性质的时间复杂度提供了更好的控制。随后，我们转向 $\mathbb{R}^d$ 中的半空间——这是性质测试和学习理论中的一个基本类。我们研究了在加法误差 $ε$ 内近似输入函数到最近半空间的距离问题。对于无分布距离近似问题，已知算法实现了查询复杂度 $O(d/ε^2)$，但所需时间为 $\tildeΘ(1/ε^d)$。我们为这一差距提供了细粒度论证：假设 $k$-SUM 猜想，任何算法的运行时间必须为 $Ω(1/ε^{d/2})$。这一细粒度下界为自然且被广泛研究的（容错）测试问题中查询复杂度与时间复杂度之间的可证明分离提供了依据。我们还证明了在标准高斯分布下，任何统计查询（SQ）算法如果被回答的查询带有高达 $ε^{Ω(d)}$ 的加法误差，则需要 $(1/ε)^{Ω(d)}$ 次查询，这揭示了即使在分布特定设置中也存在基本障碍。