  The quantum Hall states at filling factors $\nu=5/2$ and $7/2$ are expected
to have Abelian charge $e/2$ quasiparticles and non-Abelian charge $e/4$
quasiparticles. The non-Abelian statistics of the latter has been predicted to
display a striking interferometric signature, the even-odd effect. By measuring
resistance oscillations as a function of magnetic field in Fabry-P\'erot
interferometers using new high purity heterostructures, we for the first time
report experimental evidence for the non-Abelian nature of excitations at
$\nu=7/2$. At both $\nu=5/2$ and $7/2$ we also examine, for the first time, the
fermion parity, a topological quantum number of an even number of non-Abelian
quasiparticles. The phase of observed $e/4$ oscillations is reproducible and
stable over long times (hours) near both filling factors, indicating stability
of the fermion parity. At both fractions, when phase fluctuations are observed,
they are predominantly $\pi$ phase flips, consistent with either fermion parity
change or change in the number of the enclosed $e/4$ quasiparticles. We also
examine lower-frequency oscillations attributable to Abelian interference
processes in both states. Taken together, these results constitute new evidence
for the non-Abelian nature of $e/4$ quasiparticles; the observed life-time of
their combined fermion parity further strengthens the case for their utility
for topological quantum computation.
