  We discuss how strongly interacting higher-order symmetry protected
topological (HOSPT) phases can be characterized from the entanglement
perspective: First, we introduce a topological many-body invariant which
reveals the non-commutative algebra between flux operator and $C_n$ rotations.
We argue that this invariant denotes the angular momentum carried by the
instanton which is closely related to the discrete Wen-Zee response and
fractional corner charge. Second, we define a new entanglement property, dubbed
`higher-order entanglement', to scrutinize and differentiate various
higher-order topological phases from a hierarchical sequence of the
entanglement structure. We support our claims by numerically studying a
super-lattice Bose-Hubbard model that exhibits different HOSPT phases.
