# Improved Approximate Degree Bounds For k-distinctness

An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element Distinctness) is well-understood, there is a polynomial gap between the known upper and lower bounds for all constants k>2. Specifically, the best known upper bound is O(N^(3/4)-1/(2^k+2-4)) (Belovs, FOCS 2012), while the best known lower bound for k >= 2 is Omega(N^2/3 + N^(3/4)-1/(2k)) (Aaronson and Shi, J. ACM 2004; Bun, Kothari, and Thaler, STOC 2018). For any constant k >= 4, we improve the lower bound to Omega(N^(3/4)-1/(4k)). This yields, for example, the first proof that 4-distinctness is strictly harder than Element Distinctness. Our lower bound applies more generally to approximate degree. As a secondary result, we give a simple construction of an approximating polynomial of degree O(N^3/4) that applies whenever k <= polylog(N).

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