  We obtain new lower bounds for the first non-zero eigenvalue of the scalar
sub-Laplacian for 3-Sasaki metrics, improving Lichnerowicz-Obata type estimates
by Ivanov et al. The limiting eigenspace is fully decribed in terms of the
automorphism algebra. Our results can be thought of as an analogue of the
Lichnerowicz-Matsushima estimate for K\"ahler-Einstein metrics. In dimension 7,
if the automorphism algebra is non-vanishing, we also compute the second
eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In
addition, for all metrics in the canonical variation of the 3-Sasaki metric we
give a lower bound for the spectrum of the Riemannian Laplace operator,
depending only on scalar curvature and dimension. We also strengthen a result
pertaining to the growth rate of harmonic functions, due to Conlon, Hein and
Sun, in the case of hyperk\"ahler cones. In this setup we also describe the
space of holomorphic functions.
