Abstract: A classical problem in number theory is to estimate the size of the least quadratic non-residue modulo a prime. On the other hand, a Fourier optimization problem is to optimize a given quantity, given restrictions on a function and its Fourier transform. We will discuss these two concepts and a framework that connects them, and explore how this connection leads to subtle, but conceptually interesting, improvements on the best current asymptotic bounds on the former (under the Generalized Riemann Hypothesis), given by Lamzouri, Li, and Soundararajan. Based on joint work with Emanuel Carneiro, Micah Milinovich, and Antonio Ramos.