  Precise thermometry for quantum systems is important to the development of
new technology, and understanding the ultimate limits to precision presents a
fundamental challenge. It is well known that optimal thermometry requires
projective measurements of the total energy of the sample. However, this is
infeasible in even moderately-sized systems, where realistic energy
measurements will necessarily involve some coarse graining. Here, we explore
the precision limits for temperature estimation when only coarse-grained
measurements are available. Utilizing tools from signal processing, we derive
the structure of optimal coarse-grained measurements and find that good
temperature estimates can generally be attained even with a small number of
outcomes. We apply our results to many-body systems and nonequilibrium
thermometry. For the former, we focus on interacting spin lattices, both at and
away from criticality, and find that the Fisher-information scaling with system
size is unchanged after coarse-graining. For the latter, we consider a probe of
given dimension interacting with the sample, followed by a measurement of the
probe. We derive an upper bound on arbitrary, nonequilibrium strategies for
such probe-based thermometry and illustrate it for thermometry on a
Bose-Einstein condensate using an atomic quantum-dot probe.
