  We characterize the limiting distribution of Rudin-Shapiro polynomials,
showing that, normalized, their values become uniformly distributed in the
disc. This resolves conjectures of Saffari and Montgomery. Our proof proceeds
by relating the polynomials' distribution to that of a product of weakly
dependent random matrices, which we analyze using the representation theory of
$SU(2)$. Our approach leads us to a non-commutative analogue of the classical
central limit theorem of Salem and Zygmund, which may be of independent
interest.
