A $K$-multi-collision-resistant hash function ($K$-MCRH) is a shrinking keyed function for which it is computationally infeasible to find $K$ distinct inputs that map to the same output under a randomly chosen hash key; the case $K = 2$ coincides with the standard definition of collision-resistant hash function (CRH).
A natural question is whether $K$-MCRH implies CRH for $K \geq 3$, as noted by Komargodski, Naor, and Yogev (EUROCRYPT 2018) and also by Jain, Li, Robere, and Xun (FOCS 2024).

We resolve this question for all constant $K$, showing that there is no black-box construction of $K$-MCRH from $(K + 1)$-MCRH for all constant $K \geq 2$.
We also show that there is no black-box construction of distributional CRH (which is another relaxation of CRH) from 3-MCRH, answering an open question posed by Komargodski and Yogev (CRYPTO 2018) and also by Berman, Degwekar, Rothblum, and Vasudevan (EUROCRYPT 2018).
Besides applications in cryptography, our separation also implies black-box separations between TFNP search problems, which are related to problems in proof complexity and other areas.