  The quest for universal signatures of topological phases is fundamentally
important since these properties are robust to variations in system-specific
details. Here we present general results for the response of quantum Hall
states to points of singular curvature in real space. Such topological
singularities may be realized, for instance, at the vertices of a cube, the
apex of a cone, etc. We find, using continuum analytical methods, that the
point of curvature binds an excess fractional charge. In addition, sequences of
states split away, energetically, from the degenerate bulk Landau levels.
Importantly, these inter-Landau level states are bound to the topological
singularity and have energies that are $\emph{universal}$ functions of bulk
parameters and the curvature. Remarkably, our exact diagonalization of lattice
tight-binding models on closed manifolds shows that these results continue to
hold even when lattice effects are significant, where the applicability of
continuum techniques could not have been justified a priori. Moreover, we
propose how these states may be readily experimentally actualized. An immediate
technological implication of these results is that these inter-Landau level
states, being as they are $\emph{both}$ energetically and spatially isolated
quantum states, are promising candidates for constructing qubits for quantum
computation.
