  We study the long time dynamics of the Schr\"odinger equation on Zoll
manifolds. We establish criteria under which the solutions of the Schr\"odinger
equation can or cannot concentrate on a given closed geodesic. As an
application, we derive some results on the set of semiclassical measures for
eigenfunctions of Schr\"odinger operators: we prove that adding a potential to
the Laplacian on the sphere results on the existence of geodesics $\gamma$ such
that $\delta_\gamma$ cannot be obtained as a semiclassical measure for some
sequence of eigenfunctions. We also show that the same phenomenon occurs for
the free Laplacian on certain Zoll surfaces.
