We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition \textbarS⟩ of its elements. One can think of \textbarS⟩=∑i∈S\textbari⟩/\textbarS\textbar−−−√ as the quantum version of a uniformly random sample over S, as in the classical analysis of the ‘‘coupon collector problem.&⋕39;&⋕39; We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n−k=O(1) missing elements then O(k) copies of \textbarS⟩ suffice, in contrast to the Θ(klogk) random samples needed by a classical coupon collector. On the other hand, if n−k=Ω(k), then Ω(klogk) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through \textbarS⟩. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.