This question is in a sense the converse of Will quantum computers out-scale classical computers at P-problems?. We know that there are oracle problems (e.g. unstructured search) for which we can prove that quantum computers can only give a fixed polynomial speedup (in that case quadratic) for the number of oracle consultations. Are there any problems where we can prove that quantum computers can only give a constant speedup? Or even rule out a constant greater than 1?
(I'm excluding oracle problems where there's no quantum speedup for the trivial reason that the classical time complexity is already constant!)