  We derive general results relating revivals in the dynamics of quantum
many-body systems to the entanglement properties of energy eigenstates. For a
D-dimensional lattice system of N sites initialized in a low-entangled and
short-range correlated state, our results show that a perfect revival of the
state after a time at most poly(N) implies the existence of "quantum many-body
scars", whose number grows at least as the square root of N up to
poly-logarithmic factors. These are energy eigenstates with energies placed in
an equally-spaced ladder and with R\'enyi entanglement entropy scaling as
log(N) plus an area law term for any region of the lattice. This shows that
quantum many-body scars are a necessary condition for revivals, independent of
particularities of the Hamiltonian leading to them. We also present results for
approximate revivals, for revivals of expectation values of observables and
prove that the duration of revivals of states has to become vanishingly short
with increasing system size.
