  We study the influence that collective transition phenomena have on quantum
metrological protocols. The single spherical quantum spin (SQS) serves as
stereotypical toy model that allows analytical insights on a mean-field level.
First, we focus on equilibrium quantum criticality in the SQS and obtain the
quantum Fisher information analytically, which is associated with the minimum
lower bound for the precision of estimation of the parameter driving the phase
transition. We compare it with the Fisher information for a specific
experimental scenario where photoncounting-like measurements are employed. We
find how quantum criticality and squeezing are useful resources in the
metrological scenario. Second, we obtain the quantum Fisher information for the
out of equilibrium transition in the dissipative non-equilibrium steady state
(NESS), and investigate how the presence of dissipation affects the parameter
estimation. In this scenario, it is known that the critical point is shifted by
an amount which depends on the dissipation rate. This is used here to design
high precision protocols for a whole range of the transition-driving parameter
in the ordered phase. In fact, for certain values of the parameter being
estimated, dissipation may be used to obtain higher precision when compared to
the equilibrium scenario.
