Cosmic microwave background studies of non-Gaussianity involving higher order multispectra can distinguish between early universe theories that predict nearly identical power spectra. However, the recovery of higher order multispectra is difficult from realistic data due to their complex response to inhomogeneous noise and partial sky coverage, which are often difficult to model analytically. A traditional alternative is to use one-point cumulants of various orders, which collapse the information present in a multispectrum to one number. The disadvantage of such a radical compression of the data is a loss of information as to the source of the statistical behaviour. A recent study by Munshi & Heavens has shown how to define the skew spectrum (the power spectra of a certain cubic field, related to the bispectrum) in an optimal way and how to estimate it from realistic data. The skew spectrum retains some of the information from the full configuration dependence of the bispectrum and can contain all the information on non-Gaussianity. In this study, we extend the results of the skew spectrum to the case of two degenerate power spectra related to the trispectrum. We also explore the relationship between these power spectra and cumulant correlators previously used to study non-Gaussianity in projected galaxy surveys or weak-lensing surveys. We construct nearly optimal estimators for quick tests and generalize them to estimators which can handle realistic data with all their complexity in a completely optimal manner. Possible generalizations for arbitrary order are also discussed.