The closest pair problem is a fundamental problem of computational geometry: given a set of (n) points in a (d)-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in (O(nlog n)) time in constant dimensions (i.e., when (d=O(1))). This paper asks and answers the question of the problem’s quantum time complexity. Specifically, we give an (\tilde{O}(n^{2/3})) algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In (\mathrm{polylog}(n)) dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover’s algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover’s algorithm is optimal for CNF-SAT when the clause width is large. We show that the na"{i}ve Grover approach to closest pair in higher dimensions is optimal up to an (n^{o(1)}) factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.