  It has long been known that in the usual black-box model, one cannot get
super-polynomial quantum speedups without some promise on the inputs. In this
paper, we examine certain types of symmetric promises, and show that they also
cannot give rise to super-polynomial quantum speedups. We conclude that
exponential quantum speedups only occur given "structured" promises on the
input.
  Specifically, we show that there is a polynomial relationship of degree $12$
between $D(f)$ and $Q(f)$ for any function $f$ defined on permutations
(elements of $\{0,1,\dots, M-1\}^n$ in which each alphabet element occurs
exactly once). We generalize this result to all functions $f$ defined on orbits
of the symmetric group action $S_n$ (which acts on an element of $\{0,1,\dots,
M-1\}^n$ by permuting its entries). We also show that when $M$ is constant, any
function $f$ defined on a "symmetric set" - one invariant under $S_n$ -
satisfies $R(f)=O(Q(f)^{12(M-1)})$.
