  We consider the clock game-a task formulated in the framework of quantum
information theory-that can be used to improve the existing schemes of
quantum-enhanced telescopy. The problem of learning when a stellar photon
reaches a telescope is translated into an abstract game, which we call the
clock game. A winning strategy is provided that involves performing a quantum
non-demolition measurement that verifies which stellar spatio-temporal modes
are occupied by a photon without disturbing the phase information. We prove
tight lower bounds on the entanglement cost needed to win the clock game, with
the amount of necessary entangled bits equaling the number of time-bins being
distinguished. This lower bound on the entanglement cost applies to any
telescopy protocol that aims to non-destructively extract the time-bin
information of an incident photon through local measurements, and our result
implies that the protocol of Khabiboulline et al. [\text{Phys. Rev. Lett.} 123,
70504 (2019)] is optimal in terms of entanglement consumption. The full task of
the phase extraction is also considered, and we show that the quantum Fisher
information of the stellar phase can be achieved by local measurements and
shared entanglement without the necessity of nonlinear optical operations. The
optimal phase measurement is achieved asymptotically with increasing number of
ancilla qubits, whereas a single qubit pair is required if nonlinear operations
are allowed.
