Clocks are among the most precise measurement devices ever built, but like anything else, they are bound by the laws of thermodynamics. As a consequence, all clocks are inherently subject to noise and thus cannot be infinitely precise. We examine these limitations and the thermodynamic cost of running a clock. From minimal assumptions, we derive that processes driving a clock must be irreversible. In the simplest memory-less thermodynamic setting, this leads to exponential decay. Under the assumption of a fixed decay rate $\Gamma$, we explore what types of quantum clocks can be built using an exponentially decaying process to generate the ticks of the clock, but with a clockwork that is otherwise unconstrained. Using the average number of ticks $N$ until a clock is off by one tick as the measure for its accuracy, and the inverse average tick time as the measure for its resolution $R$, we show that any increase in accuracy ultimately comes at the cost of resolution, subject to the bound $N\leq \Gamma^2/R^2$. With a periodic process in the clockwork to concentrate the decay event probability to specific times, we can build clocks that approach this optimal accuracy-resolution trade-off. Based on a quantum clock from Schwarzhans et al. [1], that uses as its only resource out of equilibrium heat baths, we design a clock that we conjecture to reach the scaling $N\sim \text{const.}/R^2$ asymptotically. In this example, the entropy production grows linearly with the accuracy, confirming the thermodynamic limitation of the clock performance.

[1] Emanuel Schwarzhans, Maximilian P. E. Lock, Paul Erker, Nicolai Friis, and Marcus Huber. Autonomous Temporal Probability Concentration: Clockworks and the Second Law of Thermodynamics. RX, 11(1):011046, 2021. doi: 10.1103/PhysRevX.11.011046. URL https://link.aps.org/doi/10.1103/PhysRevX.11.011046.