  We investigate lower asymptotic bounds of number variances for invariant
locally square-integrable random measures on Euclidean and real hyperbolic
spaces. In the Euclidean case we show that there are subsequences of radii for
which the number variance grows at least as fast as the volume of the boundary
of Euclidean balls, generalizing a classical result of Beck. With regards to
real hyperbolic spaces we prove that random measures are never geometrically
hyperuniform and if the random measure admits non-trivial complementary series
diffraction, then it is hyperfluctuating. Moreover, we define spectral
hyperuniformity and stealth of random measures on real hyperbolic spaces in
terms of vanishing of the complementary series diffraction and sub-Poissonian
decay of the principal series diffraction around the Harish-Chandra
$\Xi$-function.
