  We consider the problem of testing uniformity on high-dimensional unit
spheres. We are primarily interested in non-null issues. We show that
rotationally symmetric alternatives lead to two Local Asymptotic Normality
(LAN) structures. The first one is for fixed modal location $\theta$ and allows
to derive locally asymptotically most powerful tests under specified $\theta$.
The second one, that addresses the Fisher-von Mises-Langevin (FvML) case,
relates to the unspecified-$\theta$ problem and shows that the high-dimensional
Rayleigh test is locally asymptotically most powerful invariant. Under mild
assumptions, we derive the asymptotic non-null distribution of this test, which
allows to extend away from the FvML case the asymptotic powers obtained there
from Le Cam's third lemma. Throughout, we allow the dimension $p$ to go to
infinity in an arbitrary way as a function of the sample size $n$. Some of our
results also strengthen the local optimality properties of the Rayleigh test in
low dimensions. We perform a Monte Carlo study to illustrate our asymptotic
results. Finally, we treat an application related to testing for sphericity in
high dimensions.
