  The Berry heuristic has been a long standing \emph{ansatz} about the high
energy (i.e. large eigenvalues) behaviour of eigenfunctions (see Berry 1977).
Roughly speaking, it states that under some generic boundary conditions, these
eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to
infinity. Our aim in this paper is to make this statement quantitative and to
establish some rigorous bounds on the distance to Gaussianity, focussing on the
hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator
on the normalized $d$-dimensional sphere - also known as spherical harmonics).
Some applications to non-Gaussian models are also discussed.
