Trigonometric QSP

There exist angles $\omega\in\mathbb{R}$ and $\boldsymbol{\theta},\boldsymbol{\phi}\in\mathbb{R}^{L+1}$ such that $$ U_{\omega,\boldsymbol{\theta},\boldsymbol{\phi}}^{L}(x) = R_Z(\omega)\,R_Y(\theta_0)R_Z(\phi_0) \prod_{j=1}^{L}\bigl(R_Z(x)\,R_Y(\theta_j)R_Z(\phi_j)\bigr) = \begin{bmatrix} P(x) & -Q(x)\\ Q^*(x) & P^*(x) \end{bmatrix} $$ if and only if $P,Q\in\mathbb{C}[e^{ix/2},e^{-ix/2}]$ satisfy $\deg(P)\leq L$, $\deg(Q)\leq L$, $P$ and $Q$ have parity $L\bmod 2$, and $|P(x)|^2+|Q(x)|^2=1$ for all $x\in\mathbb{R}$.