  Quantum signal processing (QSP) is a highly successful algorithmic primitive
in quantum computing which leads to conceptually simple and efficient quantum
algorithms using the block-encoding framework of quantum linear algebra.
Multivariate variants of quantum signal processing (MQSP) could be a valuable
tool in extending earlier results via implementing multivariate (matrix)
polynomials. However, MQSP remains much less understood than its single-variate
version lacking a clear characterization of "achievable" multivariate
polynomials. We show that Haah's characterization of general univariate QSP can
be extended to homogeneous bivariate (commuting) quantum signal processing. We
also show a similar result for an alternative inhomogeneous variant when the
degree in one of the variables is at most 1, but construct a counterexample
where both variables have degree 2, which in turn refutes an earlier
characterization proposed / conjectured by Rossi and Chuang for a related
restricted class of MQSP. Finally, we describe homogeneous multivariate
(non-commuting) QSP variants that break away from the earlier two-dimensional
treatment limited by its reliance on Jordan-like decompositions, and might
ultimately lead to the development of novel quantum algorithms.
