  We present results on the properties of neon emission in $z\sim2$
star-forming galaxies drawn from the MOSFIRE Deep Evolution Field (MOSDEF)
survey. Doubly-ionized neon ([NeIII]3869) is detected at $\geq3\sigma$ in 61
galaxies, representing $\sim$25% of the MOSDEF sample with H$\alpha$, H$\beta$,
and [OIII]$5007$ detections at similar redshifts. We consider the neon
emission-line properties of both individual galaxies with [NeIII]3869
detections and composite $z\sim2$ spectra binned by stellar mass. With no
requirement of [NeIII]3869 detection, the latter provide a more representative
picture of neon emission-line properties in the MOSDEF sample. The
[NeIII]3869/[OII]3727 ratio (Ne3O2) is anti-correlated with stellar mass in
$z\sim2$ galaxies, as expected based on the mass-metallicity relation. It is
also positively correlated with the [OIII]$5007$/[OII]$3727$ ratio (O32), but
$z\sim2$ line ratios are offset towards higher Ne3O2 at fixed O32, compared
with both local star-forming galaxies and individual H~II regions. Despite the
offset towards higher Ne3O2 at fixed O32 at $z\sim2$, biases in inferred
Ne3O2-based metallicity are small. Accordingly, Ne3O2 may serve as an important
metallicity indicator deep into the reionization epoch. Analyzing additional
rest-optical line ratios including [NeIII]$3869$/[OIII]$5007$ (Ne3O3) and
[OIII]$5007$/H$\beta$ (O3H$\beta$), we conclude that the nebular emission-line
ratios of $z\sim2$ star-forming galaxies suggest a harder ionizing spectrum
(lower stellar metallicity, i.e., Fe/H) at fixed gas-phase oxygen abundance,
compared to systems at $z\sim0$. These new results based on neon lend support
to the physical picture painted by oxygen, nitrogen, hydrogen, and sulfur
emission, of an ionized ISM in high-redshift star-forming galaxies irradiated
by chemically young, $\alpha$-enhanced massive stars.
