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This paper considers the query complexity of the functions in the familyF_{N,M} of N-variable Boolean functions with onset size M, i.e., the number ofinputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumedwithout loss of generality because of the symmetry of function values, 0 and 1.Our main results are as follows: (1) There is a super-linear gap between theaverage-case and worst-case quantum query complexities over F_{N,M} for acertain range of M. (2) There is no super-linear gap between the average-caseand worst-case randomized query complexities over F_{N,M} for every M. (3) Forevery M bounded by a polynomial in N, any function in F_{N,M} has quantum querycomplexity Theta (sqrt{N}). (4) For every M=O(2^{cN}) with an arbitrary largeconstant c<1, any function in F_{N,M} has randomized query complexity Omega(N).

Average/Worst-Case Gap of Quantum Query Complexities by On-Set Size