A $k$-collision for a compressing hash function $H$ is a set of $k$ distinct inputs that all map to the same output. In this work, we show that for any constant $k$, $Thetaleft(N^{frac{1}{2}(1-frac{1}{2^k-1})}right)$ quantum queries are both necessary and sufficient to achieve a $k$-collision with constant probability. This improves on both the best prior upper bound (Hosoyamada et al., ASIACRYPT 2017) and provides the first non-trivial lower bound, completely resolving the problem.
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