Motivated by the quantum unique ergodicity conjecture, we examine semiclassical measures for Laplace eigenfunctions on compact hyperbolic n-manifolds. We prove their support must contain the cosphere bundle of a compact immersed totally geodesic submanifold.The proof adapts a 2-dimensional argument by Dyatlov and Jin to higher dimensions using Cohen's higher-dimensional fractal uncertainty principle. To explain some of the main proof ideas, we will present work on the toy model of quantum cat maps.This is joint work with Nick Miller.