  A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16)
shows that any bounded quadratic polynomial can be computed exactly in
expectation by a 1-query algorithm up to a universal multiplicative factor
related to the famous Grothendieck constant. Here we show that such a result
does not generalize to quartic polynomials and 2-query algorithms, even when we
allow for additive approximations. We also show that the additive approximation
implied by their result is tight for bounded bilinear forms, which gives a new
characterization of the Grothendieck constant in terms of 1-query quantum
algorithms. Along the way we provide reformulations of the completely bounded
norm of a form, and its dual norm.
