  Phylogenetic networks are a generalization of phylogenetic trees to
leaf-labeled directed acyclic graphs that represent ancestral relationships
between species whose past includes non-tree-like events such as hybridization
and horizontal gene transfer. Indeed, each phylogenetic network embeds a
collection of phylogenetic trees. Referring to the collection of trees that a
given phylogenetic network $N$ embeds as the display set of $N$, several
questions in the context of the display set of $N$ have recently been analyzed.
For example, the widely studied Tree-Containment problem asks if a given
phylogenetic tree is contained in the display set of a given network. The focus
of this paper are two questions that naturally arise in comparing the display
sets of two phylogenetic networks. First, we analyze the problem of deciding if
the display sets of two phylogenetic networks have a tree in common.
Surprisingly, this problem turns out to be NP-complete even for two temporal
normal networks. Second, we investigate the question of whether or not the
display sets of two phylogenetic networks are equal. While we recently showed
that this problem is polynomial-time solvable for a normal and a tree-child
network, it is computationally hard in the general case. In establishing
hardness, we show that the problem is contained in the second level of the
polynomial-time hierarchy. Specifically, it is $\Pi_2^P$-complete. Along the
way, we show that two other problems are also $\Pi_2^P$-complete, one of which
being a generalization of Tree-Containment.
