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Multiple pulsar-timing-array collaborations have reported strong evidence for the existence of a gravitational-wave background. We study physical implications of this signal for cosmology, assuming that it is attributed to scalar-induced gravitational waves. By incorporating primordial non-Gaussianity f_{NL}, we specifically examine the nature of primordial curvature perturbations and primordial black holes. We find that the signal allows for a primordial non-Gaussianity f_{NL} in the range of −4.1≲f_{NL}≲4.1 (68% confidence intervals) and a mass range for primordial black holes m_{pbh} spanning from ∼10^{−5}M_{⊙} to ∼10^{−2}M_{⊙}. Furthermore, we find that the signal favors a negative non-Gaussianity, which can suppress the abundance of primordial black holes. We also demonstrate that the anisotropies of scalar-induced gravitational waves serve as a powerful tool to probe the non-Gaussianity f_{NL}. We conduct a comprehensive analysis of the angular power spectrum within the nano-Hertz band. Looking ahead, we anticipate that future projects, such as the Square Kilometre Array, will have the potential to measure these anisotropies and provide further insights into the primordial universe.