Beyond the amplitude of primordial non-Gaussianity lie its higher moments and its running: the trispectrum amplitudes g_NL and tau_NL from the four-point function, and the scale dependence n_fNL describing how f_NL drifts with wavenumber. Planck constrains g_NL local = (-5.8 +/- 6.5) x 10^4 and tau_NL below about 2800 at 95 percent confidence, both consistent with zero, while the running of f_NL is essentially unconstrained by current data. These observables matter because they encode consistency relations: in broad classes of models the trispectrum is tied to the square of the bispectrum (the Suyama-Yamaguchi inequality tau_NL greater than or equal to (6 f_NL/5)^2), and violations or specific hierarchies among f_NL, g_NL, and tau_NL discriminate mechanisms that identical f_NL values cannot.
The ΛCDM predicament mirrors the f_NL story at higher order with even less constraint: single-field slow-roll predicts utterly undetectable values, multi-field models populate the whole accessible plane, and the running n_fNL is a free function of the unspecified potential landscape. The data are weakest exactly where models differ most: a measured g_NL hierarchy or a detected running would be enormously discriminating, but no standard-model variant stakes a falsifiable claim about what those values must be. The higher-point sector is a second sensitivity race with no committed standard-model entry.
The standing is forward-looking: CMB-S4 will improve trispectrum constraints by an order of magnitude, SPHEREx targets both f_NL and its running through multi-tracer scale-dependent bias, and 21-cm surveys ultimately offer the largest mode counts. Any framework that fixes the full hierarchy of cumulants and their scale dependence from one mechanism, rather than from adjustable potentials, will face an increasingly sharp multi-observable test this decade.