  We study holographic entropy inequalities and their structural properties by
making use of a judicious grouping of terms into certain multipartite
information quantities. This allows us to recast cumbersome entropic
expressions into much simpler ones which share interestingly rigid structures.
By performing a systematic search over some of these structures, we are able to
discover more than 300 novel entropy inequalities for six parties, thereby
demonstrating that these recastings provide a fruitful generating technique for
uncovering new holographic entropy inequalities. In attempting to interpret the
corresponding sign-definite quantities as correlation measures, we also obtain
a no-go result: the superbalance property of holographic entropy inequalities
turns out to preclude them from being monotonic under partial tracing. In the
process, we also comment on the geometrical significance of multipartite
information quantities and present various structural relations amongst them.
