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We give a lower bound of 茂戮驴(n(d茂戮驴 1)/2) on the quantum query complexity for finding a fixed point of a discrete Brouwer function over grid [n]d. Our lower bound is nearly tight, as Grover Search can be used to find a fixed point with O(nd/2) quantum queries. Our result establishes a nearly tight bound for the computation of d-dimensional approximate Brouwer fixed points defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. It can be extended to the quantum model for Sperner's Lemma in any dimensions: The quantum query complexity of finding a panchromatic cell in a Sperner coloring of a triangulation of a d-dimensional simplex with ndcells is 茂戮驴(n(d茂戮驴 1)/2). For d= 2, this result improves the bound of 茂戮驴(n1/4) of Friedl, Ivanyos, Santha, and Verhoeven.More significantly, our result provides a quantum separation of local search and fixed point computation over [n]d, for d茂戮驴 4. Aaronson's local search algorithm for grid [n]d, using Aldous Sampling and Grover Search, makes O(nd/3) quantum queries. Thus, the quantum query model over [n]dfor d茂戮驴 4 strictly separates these two fundamental search problems.