  The Gottesman-Knill theorem holds that operations from the Clifford group,
when combined with preparation and detection of qubit states in the
computational basis, are insufficient for universal quantum computation.
Indeed, any measurement results in such a system could be reproduced within a
local hidden variable theory, so that there is no need for a quantum mechanical
explanation and therefore no possibility of quantum speedup. Unfortunately,
Clifford operations are precisely the ones available through braiding and
measurement in systems supporting non-Abelian Majorana zero modes, which are
otherwise an excellent candidate for topologically protected quantum
computation. In order to move beyond the classically simulable subspace, an
additional phase gate is required. This phase gate allows the system to violate
the Bell-like CHSH inequality that would constrain a local hidden variable
theory. In this article, we both demonstrate the procedure for measuring Bell
violations in Majorana systems and introduce a new type of phase gate for the
already existing semiconductor-based Majorana wire systems. We conclude with an
experimentally feasible schematic combining the two, which should potentially
lead to the demonstration of Bell violation in a Majorana experiment in the
near future. Our work also naturally leads to a well-defined platform for
universal fault-tolerant quantum computation using Majorana zero modes, which
we describe.
