  Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced
for merge trees and multiparameter persistence modules as more sensitive
variations of the respective interleaving distances ($p=\infty$). It is
well-known that computing the interleaving distance is NP-hard in both cases.
We extend this result by showing that computing the $p$-presentation distance
is NP-hard for all $p\in [1,\infty)$ for both merge trees and $t$-parameter
persistence modules for any $t\geq 2$. Though the details differ, both proofs
follow the same novel strategy, suggesting that our approach can be adapted to
proving the NP-hardness of other distances based on sums or $p$-norms.
