The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube {0, 1} n , we show a lower bound of Ω 2 n/4 /n on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis' quantum adversary method, also yields a lower bound of Ω 2 n/2 /n 2 on the problem's classical randomized query complexity. This improves

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... nd simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension greater than 2. *