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It is known since the work of [AA14] that for any permutation symmetric function $f$, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity, more precisely that $R(f) = \widetilde{O}\left(Q^7(f)\right)$.
Rapid solution of problems by quantum computation
D. Deutsch and R. Jozsa · 1992
Earlier work this paper cites.
Algorithms for quantum computation: Discrete logarithms and factoring
Peter W. Shor · 1994
Earlier work this paper cites.
On the power of quantum cryptography
Daniel R. Simon · 1994
Earlier work this paper cites.
Strengths and weaknesses of quantum computing
Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani · 1997
Earlier work this paper cites.
Quantum lower bounds by polynomials
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf · 2001
Cited alongside, same era.
Quantum lower bounds for the collision and the element distinctness problems
Scott Aaronson and Yaoyun Shi · 2004
Cited alongside, same era.
Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range
Andris Ambainis · 2005
Cited alongside, same era.
Quantum lower bound for the collision problem with small range
Samuel Kutin · 2005
Cited alongside, same era.
The need for structure in quantum speedups
Scott Aaronson and Andris Ambainis · 2014
Later among the works it cites.
A note on the quantum collision and set equality problems
Mark Zhandry · 2015
Later among the works it cites.
Understanding Quantum Algorithms via Query Complexity
A. Ambainis · 2017
Later among the works it cites.
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