  One of the main topological invariants that characterizes several
topologically-ordered phases is the many-body Chern number (MBCN). Paradigmatic
examples include several fractional quantum Hall phases, which are expected to
be realized in different atomic and photonic quantum platforms in the near
future. Experimental measurement and numerical computation of this invariant is
conventionally based on the linear-response techniques which require having
access to a family of states, as a function of an external parameter, which is
not suitable for many quantum simulators. Here, we propose an ancilla-free
experimental scheme for the measurement of this invariant, without requiring
any knowledge of the Hamiltonian. Specifically, we use the statistical
correlations of randomized measurements to infer the MBCN of a wavefunction.
Remarkably, our results apply to disk-like geometries that are more amenable to
current quantum simulator architectures.
