  In this work we study genuine multipartite correlations (GMC's) in a boundary
time crystal (BTC). Boundary time crystals are nonequilibrium quantum phases of
matter in contact to an environment, for which a macroscopic fraction of the
many-body system breaks the time-translation symmetry. We analyze both (i) the
structure (orders) of GMC's among the subsystems, as well as (ii) their
build-up dynamics for an initially uncorrelated state. We find that, in the
thermodynamic limit (and only in such a limit), multipartite correlations of
all orders grow indefinitely in time in the BTC phase, further displaying a
persistent oscillatory behavior around their mean growth. The orders of the
correlations show a power-law decaying hierarchy among its $k$-partitions.
Moreover, in the long-time limit the GMC's are shown extensive with the system
size, contrasting to the subextensive scaling in the non time-crystal
(ferromagnetic) phase of the model. We also discuss the classical and quantum
nature of these correlations with basis on multipartite entanglement witnesses,
specifically, the analysis of the Quantum Fisher Information (QFI). Both GMC
and QFI are able to capture and distinguish the different phases of the model.
Our work highlights the genuine many-body properties of these peculiar
non-equilibrium phases of matter.
