  Let $M\subset \mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be a compact cmc
rotational hypersurface of the $(n+1)$-dimensional Euclidean unit sphere.
Denote by $|A|^2$ the square of the norm of the second fundamental form and
$J(f)=-\Delta f-nf-|A|^2f$ the stability or Jacobi operator. In this paper we
compute the spectra of their Laplace and Jacobi operators in terms of
eigenvalues of second order Hill's equations.
  For the minimal rotational examples, we prove that the stability index --the
numbers of negative eigenvalues of the Jacobi operator counted with
multiplicity -- is greater than $3 n+4$ and we also prove that there are at
least 2 positive eigenvalues of the Laplacian of $M$ smaller than $n$. When $H$
is not zero, we have that every non-flat CMC rotational immersion is generated
by rotating a planar profile curve along a geodesic called the axis of
rotation. Let $m$ be the number of points where the maximal distance from this
profile curve to the origin is achieved (we assume that the coordinates of the
plane containing the profile curve has been set up so that the axis of rotation
goes through the origin). Let $l$ be be the wrapping number of the profile
curve. We show that the number of negative eigenvalues of the operator $J$
counted with multiplicity is at least $(2l-1)n+(2m-1)$. This result was proven
for the case $n=2$ by Rossman and Sultana. They called $m$ the number of bulges
or the number of necks. We will slightly change the definition of $l$ to
include immersed examples that contain the axis of rotation.
