  Quantum particles on a lattice with competing long-range interactions are
ubiquitous in physics. Transition metal oxides, layered molecular crystals and
trapped ion arrays are a few examples out of many. In the strongly interacting
regime, these systems often exhibit a rich variety of quantum many-body ground
states that challenge theory. The emergence of transition metal dichalcogenide
moir\'e heterostructures provides a highly controllable platform to study
long-range electronic correlations. Here we report an observation of nearly
two-dozen correlated insulating states at fractional fillings of a
WSe$_{2}$/WS$_{2}$ moir\'e heterostructure. The discovery is enabled by a new
optical sensing technique that is built on the sensitivity to dielectric
environment of the exciton excited states in single-layer semiconductor
WSe$_{2}$. The cascade of insulating states exhibits an energy ordering which
is nearly symmetric about filling factor of half electron (or hole) per
superlattice site. We propose a series of charge-ordered states at commensurate
filling fractions that range from generalized Wigner crystals to charge density
waves. Our study lays the groundwork for utilizing moir\'e superlattices to
simulate a wealth of quantum many-body problems that are described by the
two-dimensional t-V model or spin models with long-range charge-charge and
exchange interactions.
