  Quantum many-body scar states are many-body states with finite energy density
in non-integrable models that do not obey the eigenstate thermalization
hypothesis. Recent works have revealed "towers" of scar states that are exactly
known and are equally spaced in energy, specifically in the AKLT model, the
spin-1 XY model, and a spin-1/2 model that conserves number of domain walls. We
provide a common framework to understand and prove known exact towers of scars
in these systems, by evaluating the commutator of the Hamiltonian and a ladder
operator. In particular we provide a simple proof of the scar towers in the
integer-spin 1d AKLT models by studying two-site spin projectors. Through this
picture we deduce a family of Hamiltonians that share the scar tower with the
AKLT model, and also find common parent Hamiltonians for the AKLT and XY model
scars. We also introduce new towers of exact states, organized in a "pyramid"
structure, in the spin-1/2 model through successive application of a non-local
ladder operator.
