  Bell's inequality is established based on local realism. The violation of
Bell's inequality by quantum mechanics implies either locality or realism or
both are untenable. Leggett's inequality is derived based on nonlocal realism.
The violation of Leggett's inequality implies that quantum mechanics is neither
local realistic nor nonlocal realistic. The incompatibility of nonlocal realism
and quantum mechanics has been urrently confirmed by photon experiments. In our
work, we propose to test Leggett's inequality using the Aharonov-Casher effect.
In our scheme, four entangled particles emitted from two sources manifest a
two-qubit-typed correlation that may result in the violation of the Leggett
inequality, while satisfying the no-signaling condition for spacelike
separation. Our scheme is tolerant to some local inaccuracies due to the
topological nature of the Aharonov-Casher phase. The experimental
implementation of our scheme can be possibly realized by a calcium atomic
polarization interferometer experiment.
