  In a recent letter [N. V. Krishnendu et al., PRL 119, 091101 (2017)] we
explored the possibility of probing the binary black hole nature of coalescing
compact binaries, by measuring their spin-induced multipole moments, observed
in advanced LIGO detectors. Coefficients characterizing the spin-induced
multipole moments of Kerr black holes are predicted by the "no-hair" conjecture
and appear in the gravitational waveforms through quadratic and higher order
spin interactions and hence can be directly measured from gravitational wave
observations. We assess the capabilities of the third-generation gravitational
wave interferometers such as Cosmic Explorer and Einstein Telescope in carrying
out such measurements and use them to test the binary black hole nature of
observed binaries. In this paper, we extend the investigations of our previous
work, by proposing to measure (a) spin-induced quadrupole effects, (b)
simultaneous measurements of spin-induced quadrupole and octupole effects, in
the context of the third-generation detectors. We find that, using
third-generation detectors the symmetric combination of coefficients associated
with spin-induced quadrupole moment of each binary component may be constrained
to a value $\leq 1.1$ while a similar combination of coefficients for
spin-induced octupole moment may be constrained to $\leq 2$, where both
combinations take the value of 1 for a binary black hole system. Further, we
consider two different binary black hole populations, as proxies of the
population that will be observed by the third generation detectors, and obtain
the resulting distribution of the spin-induced quadrupole coefficient. These
estimates suggest that third-generation detectors can accurately constrain the
first four multipole moments of the compact objects (mass, spin, quadrupole,
and octupole) facilitating a thorough probe of their black hole nature.
