  Time remains one of the least well understood concepts in physics, most
notably in quantum mechanics. A central goal is to find the fundamental limits
of measuring time. One of the main obstacles is the fact that time is not an
observable and thus has to be measured indirectly. Here we explore these
questions by introducing a model of time measurements that is complete and
autonomous. Specifically, our autonomous quantum clock consists of a system out
of thermal equilibrium --- a prerequisite for any system to function as a clock
--- powered by minimal resources, namely two thermal baths at different
temperatures. Through a detailed analysis of this specific clock model, we find
that the laws of thermodynamics dictate a trade-off between the amount of
dissipated heat and the clock's performance in terms of its accuracy and
resolution. Our results furthermore imply that a fundamental entropy production
is associated with the operation of any autonomous quantum clock, assuming that
quantum machines cannot achieve perfect efficiency at finite power. More
generally, autonomous clocks provide a natural framework for the exploration of
fundamental questions about time in quantum theory and beyond.
