We present results on the properties of neon emission in $zsim2$ star-forming galaxies drawn from the MOSFIRE Deep Evolution Field (MOSDEF) survey. Doubly-ionized neon ([NeIII]3869) is detected at $geq3sigma$ 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 $zsim2$ 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 $zsim2$ galaxies, as expected based on the mass-metallicity relation. It is also positively correlated with the [OIII]$5007$/[OII]$3727$ ratio (O32), but $zsim2$ 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 $zsim2$, 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 $zsim2$ 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 $zsim0$. 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.
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