  We consider the evolution of hypersurfaces on the unit sphere
$\mathbb{S}^{n+1}$ by smooth functions of the Weingarten map. We introduce the
notion of `quasi-ancient' solutions for flows that do not admit non-trivial,
convex, ancient solutions. Such solutions are somewhat analogous to ancient
solutions for flows such as the mean curvature flow, or 1-homogeneous flows.
The techniques presented here allow us to prove that any convex, quasi-ancient
solution of a curvature flow which satisfies a backwards in time uniform bound
on mean curvature must be stationary or a family of shrinking geodesic spheres.
The main tools are geometric, employing the maximum principle, a rigidity
result in the sphere and an Alexandrov reflection argument. We emphasize that
no homogeneity or convexity/concavity restrictions are placed on the speed,
though we do also offer a short classification proof for several such
restricted cases.
