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We show an essentially tight bound on the number of adaptively chosenstatistical queries that a computationally efficient algorithm can answeraccurately given $n$ samples from an unknown distribution. A statistical queryasks for the expectation of a predicate over the underlying distribution, andan answer to a statistical query is accurate if it is "close" to the correctexpectation over the distribution. This question was recently studied by Dworket al., who showed how to answer $\tilde{\Omega}(n^2)$ queries efficiently, andalso by Hardt and Ullman, who showed that answering $\tilde{O}(n^3)$ queries ishard. We close the gap between the two bounds and show that, under a standardhardness assumption, there is no computationally efficient algorithm that,given $n$ samples from an unknown distribution, can give valid answers to$O(n^2)$ adaptively chosen statistical queries. An implication of our resultsis that computationally efficient algorithms for answering arbitrary,adaptively chosen statistical queries may as well be differentially private.  We obtain our results using a new connection between the problem of answeringadaptively chosen statistical queries and a combinatorial object called aninteractive fingerprinting code. In order to optimize our hardness result, wegive a new Fourier-analytic approach to analyzing fingerprinting codes that issimpler, more flexible, and yields better parameters than previousconstructions.

Interactive Fingerprinting Codes and the Hardness of Preventing False Discovery