In a recent letter [textit{Phys.~Rev.~Lett.}~textbf{125}, 085001 (2020)], Dornheim textit{et al.}~have presented the first textit{ab initio} path integral Monte Carlo (PIMC) results for the nonlinear electronic density response at warm dense matter (WDM) conditions. In the present work, we extend these considerations by exploring the relation between the nonlinear response and three-/four-body correlation functions from many-body theory. In particular, this connection directly implies a comparably increased sensitivity of the nonlinear response to electronic exchange-correlation (XC) effects, which is indeed confirmed by our analysis over the entire relevant range of densities ($r_s=0.5,dots,10$) and temperatures ($theta=0.01,dots,4$). Finally, our work suggests the possibility of deliberately probing the nonlinear regime to experimentally probe three- and potentially even four-body correlation functions in WDM.
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