The statistical properties of the primordial curvature perturbations are a key ingredient of the success of the LCDM model in explaining the Universe as we observe it today.
In simplest model of inflation, initial fluctuations are Gaussian for all practical purposes, and measurements of the CMB bispectrum by the Planck satellite constrain deviations from the Gaussian regime in a part in ten thousands. On the other hand the theoretical target for the amplitude of PNG in the initial perturbations is roughly an order of magnitude smaller than what Planck has measured. We have mostly saturated the information content in the CMB, and, most likely, any further improvement will come from the late time distribution of galaxies or any other tracers of the Large Scale Structure (LSS) of the Universe. In this talk I will present new ideas to constraint PNG using the LSS. First I'll present a method to constrain PNG of the local kind using unclustered tracers of the LSS. In the limit of low noise in the data, zero bias tracers yield large improvement over standard methods, mostly due to vanishing sampling variance. We propose a simple technique to construct such a tracer, using environmental information obtained from the original sample, and validate our method with N-body simulations. Our approach is very close to optimal, and opens up the possibility to reach the theoretical target on PNG using measurements of power spectrum alone in future surveys like DESI and Euclid. In the second part of the talk I'll show how to exploit the redshift dependence, or lack thereof, of the PNG signal to disentangle it from the Gaussian terms, and derive optimal redshift weights for clustering analyses in spectroscopic surveys. An application of this new method to the Quasars in eBOSS returns one of the tightest constraint on PNG from the LSS, and shows that future eBOSS releases could become comparable with Planck precision on PNG.