  Quantum states with geometric duals are known to satisfy a stricter set of
entropy inequalities than those obeyed by general quantum systems. The set of
allowed entropies derived using the Ryu-Takayanagi (RT) formula defines the
Holographic Entropy Cone (HEC). These inequalities are no longer satisfied once
general quantum corrections are included by employing the Quantum Extremal
Surface (QES) prescription. Nevertheless, the structure of the QES formula
allows for a controlled study of how quantum contributions from bulk entropies
interplay with HEC inequalities. In this paper, we initiate an exploration of
this problem by relating bulk entropy constraints to boundary entropy
inequalities. In particular, we show that requiring the bulk entropies to
satisfy the HEC implies that the boundary entropies also satisfy the HEC.
Further, we also show that requiring the bulk entropies to obey monogamy of
mutual information (MMI) implies the boundary entropies also obey MMI.
