  We present constraints on local primordial non-Gaussianity (PNG),
parametrized through $f^{\rm loc}_{\rm NL}$, using the Sloan Digital Sky Survey
IV extended Baryon Oscillation Spectroscopic Survey Data Release 14 quasar
sample. We measure and analyze the anisotropic clustering of the quasars in
Fourier space, testing for the scale-dependent bias introduced by primordial
non-Gaussianity on large scales. We derive and employ a power spectrum
estimator using optimal weights that account for the redshift evolution of the
PNG signal. We find constraints of $-51<f^{\rm loc}_{\rm NL}<21$ at 95%
confidence level. These are amont the tightest constraints from Large Scale
Structure (LSS) data. Our redshift weighting improves the error bar by 15% in
comparison to the unweighted case. If quasars have lower response to PNG, the
constraint degrades to $-81<f^{\rm loc}_{\rm NL}<26$, with a 40% improvement
over the standard approach. We forecast that the full eBOSS dataset could reach
$\sigma_{f^{\rm loc}_{\rm NL}}\simeq 5\text{-}8$ using optimal methods and full
range of scales.
