Quantum signal processing (QSP) reduces the design of many quantum algorithms to classical problems of polynomial approximation and phase-factor computation. In this talk, I will describe how nonlinear Fourier analysis on SU(2) provides a useful framework for addressing both. First, the correspondence between QSP and the nonlinear Fourier transform (NLFT) turns phase-factor computation into an inverse NLFT problem, leading to a numerically stable inverse nonlinear fast Fourier transform with (O(n\log^2 n)) complexity and revealing connections between layer stripping, Schur’s algorithm, and structured matrix factorization.This allows fast phase factor finding for QSP protocols when a outerness condition on the polynomials is satisfied. If time permits, I will also briefly discuss constrained minimax polynomial approximation for QSP, particularly in the fully coherent regime where optimal approximants reach the boundary of the feasible set and standard discretization can leave small but consequential violations of the global boundedness constraint. We will show that a heuristic method called "Nonlinear Fourier Retraction", which uses QSP completion and inverse NLFT connections, can be used to transform a nearly feasible polynomial into an exactly feasible QSP representation. Together, these results show how nonlinear Fourier analysis can connect approximation theory, numerical algorithms, and the practical classical design of quantum signal processing.