Computational Complexity in Property Testing

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 $\tilde{\Theta}(q(n))$ and time complexity $\tilde\Omega(t(n))$. 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 $\epsilon$. For the distribution-free distance approximation problem, known algorithms achieve query complexity $O(d/\epsilon^2)$, but take time $\tilde{\Theta}(1/\epsilon^d)$. We provide a fine-grained justification for this gap: assuming the $k$-SUM conjecture, any algorithm must have running time ${\Omega}(1/\epsilon^{d/2})$. 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 $(1/\epsilon)^{\Omega(d)}$ queries if the queries are answered with additive error up to $\epsilon^{\Omega(d)}$, revealing a fundamental barrier even in the distribution-specific setting.