  We study the problem of finding $K$ collision pairs in a random function $f :
[N] \rightarrow [N]$ by using a quantum computer. We prove that the number of
queries to the function in the quantum random oracle model must increase
significantly when the size of the available memory is limited. Namely, we
demonstrate that any algorithm using $S$ qubits of memory must perform a number
$T$ of queries that satisfies the tradeoff $T^3 S \geq \Omega(K^3 N)$.
Classically, the same question has only been settled recently by Dinur
[Eurocrypt'20], who showed that the Parallel Collision Search algorithm of van
Oorschot and Wiener achieves the optimal time-space tradeoff of $T^2 S =
\Theta(K^2 N)$. Our result limits the extent to which quantum computing may
decrease this tradeoff. Our method is based on a novel application of Zhandry's
recording query technique [Crypto'19] for proving lower bounds in the
exponentially small success probability regime. As a second application, we
give a simpler proof of the time-space tradeoff $T^2 S \geq \Omega(N^3)$ for
sorting $N$ numbers on a quantum computer, which was first obtained by Klauck,
\v{S}palek and de Wolf [K\v{S}W07].
