  We prove a characterization of $t$-query quantum algorithms in terms of the
unit ball of a space of degree-$2t$ polynomials. Based on this, we obtain a
refined notion of approximate polynomial degree that equals the quantum query
complexity, answering a question of Aaronson et al. (CCC'16). Our proof is
based on a fundamental result of Christensen and Sinclair (J. Funct. Anal.,
1987) that generalizes the well-known Stinespring representation for quantum
channels to multilinear forms. Using our characterization, we show that many
polynomials of degree four are far from those coming from two-query quantum
algorithms. We also give a simple and short proof of one of the results of
Aaronson et al. showing an equivalence between one-query quantum algorithms and
bounded quadratic polynomials.
  Revision note: A mistake was found in the proof of the second result on
degree-4 polynomials far from 2-query quantum algorithms. An explanation of the
issue, a corrected proof and stronger examples are presented in work of
Escudero Guti\'errez and the second author.
