  We propose a procedure that characterizes free-fermion or interacting
multipolar higher-order topological phases via their bulk entanglement
structure. To this end, we construct nested entanglement Hamiltonians by first
applying an entanglement cut to the ordinary many-body ground state, and then
iterating the procedure by applying further entanglement cuts to the (assumed
unique) ground state of the entanglement Hamiltonian. We argue that an $n$-th
order multipolar topological phase can be characterized by the features of its
$n$-th order nested entanglement Hamiltonian e.g., degeneracy in the
entanglement spectrum. We explicitly compute nested entanglement spectra for a
set of higher-order fermionic and bosonic multipole phases and show that our
method successfully identifies such phases.
