  We develop an analytical framework to study the synchronization of a quantum
self-sustained oscillator to an external signal. Our unified description allows
us to identify the resource on which quantum synchronization relies, and to
compare quantitatively the synchronization behavior of different limit cycles
and signals. We focus on the most elementary quantum system that is able to
host a self-sustained oscillation, namely a single spin 1. Despite the spin
having no classical analogue, we first show that it can realize the van der Pol
limit cycle deep in the quantum regime, which allows us to provide an
analytical understanding to recently reported numerical results. Moving on to
the equatorial limit cycle, we then reveal the existence of an
interference-based quantum synchronization blockade and extend the classical
Arnold tongue to a snake-like split tongue. Finally, we derive the maximum
synchronization that can be achieved in the spin-1 system, and construct a
limit cycle that reaches this fundamental limit asymptotically.
