  We investigate the use of a boundary time crystals (BTCs) as sensors of AC
fields. Boundary time crystals are non-equilibrium phases of matter in contact
to an environment, for which a macroscopic fraction of the many-body system
breaks the time translation symmetry. We find an enhanced sensitivity of the
BTC when its spins are resonant with the applied AC field, as quantified by the
quantum Fisher information (QFI). The QFI dynamics in this regime is shown to
be captured by a relatively simple ansatz consisting of an initial power-law
growth and late-time exponential decay. We study the scaling of the ansatz
parameters with resources (encoding time and number of spins) and identify a
moderate quantum enhancement in the sensor performance through comparison with
classical QFI bounds. Investigating the precise source of this performance, we
find that despite of its long coherence time and multipartite correlations
(advantageous properties for quantum metrology), the entropic cost of the BTC
(which grows indefinitely in the thermodynamic limit) hinders an optimal
decoding of the AC field information. This result has implications for future
candidates of quantum sensors in open system and we hope it will encourage
future study into the role of entropy in quantum metrology.
