  Outcomes of measurements are characterized by an infinite family of
generalized uncertainties, or cumulants, which provide information beyond the
mean and variance of the observable. Here, we investigate the cumulants of a
conserved charge in a subregion with corners. We derive nonperturbative
relations for the area law, and more interestingly, the angle dependence,
showing how it is determined by geometric moments of the correlation function.
These hold for translation invariant systems under great generality, including
strongly interacting ones. We test our findings by using two-dimensional
topological quantum Hall states of bosons and fermions at both integer and
fractional fillings. We find that the odd cumulants' shape dependence differs
from the even ones. For instance, the third cumulant shows nearly universal
behavior for integer and fractional Laughlin Hall states in the lowest Landau
level. Furthermore, we examine the relation between even cumulants and the
R\'enyi entanglement entropy, where we use new results for the fractional state
at filling 1/3 to compare these quantities in the strongly interacting regime.
We discuss the implications of these findings for other systems, including
gapless Dirac fermions, and more general conformal field theories.
