The ground-state many-body electron momentum density, which can be probed by x-ray Compton scattering, holds insights into the electronic structure of materials. Comparisons between the measured so-called Compton profiles and the theoretical ones are invaluable in assessing the successes and failures of the methodology used to generate the theoretical ground-state electronic structure. Here, we present calculations of the Compton profiles of Li, Si, Cr, and Ni using the state-of-the-art QSGW method within the Questaal package compared with density functional theory (DFT), one-shot $GW$ ($G^0W^0$) predictions and with experiment. This comparison reveals significant differences between the QSGW and $G^0W^0$ methods which we attribute to the distinction between the single particle density provided by the QSGW method and the many-body density that we construct from the $G^0W^0$ theory; although in general the QSGW description of the electronic structure is superior to that of $G^0W^0$, we find the use of the many-body reduced density matrix is key to improving the agreement of the Compton profile with experiment.