  Quantum sensing enables the ultimate precision attainable in parameter
estimation. Circumstantial evidence suggests that certain organisms, most
notably migratory songbirds, also harness quantum-enhanced magnetic field
sensing via a radical-pair-based chemical compass for the precise detection of
the weak geomagnetic field. However, what underpins the acuity of such a
compass operating in a noisy biological setting, at physiological temperatures,
remains an open question. Here, we address the fundamental limits of inferring
geomagnetic field directions from radical-pair spin dynamics. Specifically, we
compare the compass precision, as derived from the directional dependence of
the radical-pair recombination yield, to the ultimate precision potentially
realisable by a quantum measurement on the spin system under steady-state
conditions. To this end, we probe the quantum Fisher information and associated
Cram\'er--Rao bound in spin models of realistic complexity, accounting for
complex inter-radical interactions, a multitude of hyperfine couplings, and
asymmetric recombination kinetics, as characteristic for the magnetosensory
protein cryptochrome. We compare several models implicated in cryptochrome
magnetoreception and unveil their optimality through the precision of
measurements ostensibly accessible to nature. Overall, the comparison provides
insight into processes honed by nature to realise optimality whilst constrained
to operating with mere reaction yields. Generally, the inference of compass
orientation from recombination yields approaches optimality in the limits of
complexity, yet plateaus short of the theoretical optimal precision bounds by
up to one or two orders of magnitude, thus underscoring the potential for
improving on design principles inherent to natural systems.
