  We study the landscapes of multi-species spherical spin glasses. Our results
determine the phase boundary for annealed trivialization of the number of
critical points, and establish its equivalence with a quenched \emph{strong
topological trivialization} property. Namely in the "trivial" regime, the
number of critical points is constant, all are well-conditioned, and all
approximate critical points are close to a true critical point. As a
consequence, we deduce that Langevin dynamics at sufficiently low temperature
has logarithmic mixing time.
  Our approach begins with the Kac--Rice formula. We derive closed form
expressions for some asymptotic determinants studied in (Ben
Arous-Bourgade-McKenna 2023, McKenna 2021), and characterize the annealed
trivialization phase by explicitly solving a suitable multi-dimensional
variational problem. To obtain more precise quenched results, we develop
general purpose techniques to avoid sub-exponential correction factors and show
non-existence of \emph{approximate} critical points. Many of the results are
new even in the $1$-species case.
