  Since the seminal work of Anderson, localisation has been recognised as a
standard mechanism allowing quantum many-body systems to escape ergodicity.
This idea acquired even more prominence in the last decade as it has been
argued that localisation -- dubbed many-body localisation (MBL) in this context
-- can sometimes survive local interactions in the presence of sufficiently
strong disorder. A conventional signature of localisation is in the statistical
properties of the spectrum -- spectral statistics -- which differ qualitatively
from those in the ergodic phase. Although features of the spectral statistics
are routinely used as numerical diagnostics for localisation, they have never
been derived from first principles in the presence of non-trivial interactions.
Here we fill this gap and provide the example of a simple class of quantum
many-body systems -- which we dub strongly localised quantum circuits -- that
are interacting, localised, and where the spectral statistics can be
characterised exactly. Furthermore, we show that these systems exhibit a
cascade of three different regimes for spectral correlations depending on the
energy scale: at small, intermediate, and large scales they behave as
disconnected patches of three decreasing sizes. We argue that these features
appear in generic MBL systems, with the difference that only at the smallest
scale they do become Poissonian.
