  We study the problem of recovering a planted hierarchy of partitions in a
network. The detectability of a single planted partition has previously been
analysed in detail and a phase transition has been identified below which the
partition cannot be detected. Here we show that, in the hierarchical setting,
there exist additional phases in which the presence of multiple consistent
partitions can either help or hinder detection. Accordingly, the detectability
limit for non-hierarchical partitions typically provides insufficient
information about the detectability of the complete hierarchical structure, as
we highlight with several constructive examples.
