Adversary Lower Bounds for the Collision and the Set Equality Problems

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We prove tight $\Omega(n^{1/3})$ lower bounds on the quantum query complexity of the Collision and the Set Equality problems, provided that the size of the alphabet is large enough. We do this using the negative-weighted adversary method. Thus, we reprove the result by Aaronson and Shi, as well as a very recent development by Zhandry.