  The understanding of particle entanglement is an important goal in the
studies of correlated quantum matter. The widely-used method of scanning
tunneling spectroscopy -- which measures the local density of states (LDOS) of
a many-body system by injecting or removing an electron from it -- is expected
to be sensitive to particle entanglement. In this paper, we systematically
investigate the relation between the particle entanglement spectrum (PES) and
the LDOS of fractional quantum Hall (FQH) states, the paradigmatic
strongly-correlated phases of electrons with topological order. Using exact
diagonalization, we show that the counting of levels in both the LDOS and PES
in the Jain sequence of FQH states can be predicted from the composite fermion
theory. We point out the differences between LDOS and PES characterization of
the bulk quasihole excitations, and we discuss the conditions under which the
LDOS counting can be mapped to that of PES. Our results affirm that tunneling
spectroscopy is a sensitive tool for identifying the nature of FQH states.
