  Twisted MoTe$_2$ homobilayers exhibit transport signatures consistent with a
fractional quantum spin Hall (FQSH) state. We describe a route to construct
topological quantum memory elements, dubbed Cheshire qudits, formed from
punching holes in such a FQSH state and using proximity-induced
superconductivity to gap out the resulting helical edge states. Cheshire qudits
encode quantum information in states that differ by a fractional topological
"Cheshire" charge that is hidden from local detection within a condensate
anyons. Control of inter-edge tunneling by gates enables both
supercurrent-based readout of a Cheshire qudit, and capacitive measurement of
the thermal entropy associated with its topological ground-space degeneracy.
Additionally, we systematically classify different types of gapped boundaries,
Cheshire qudits, and parafermionic twist defects for various Abelian and
non-Abelian candidate FQSH orders that are consistent with the transport data,
and describe experimental signatures to distinguish these orders.
