The clock synchronization problem is to determine the time difference ∆ between two spatially separated clocks. When message delivery times between the two clocks is uncertain, O(2 2n ) classical messages must be exchanged between the clocks to determine n digits of ∆. On the other hand, as we show, there exists a quantum algorithm to obtain n digits of ∆ while communicating only O(n) quantum messages.Clock synchronization is an important problem with many practical and scientific applications [1,2]. Accurate timekeeping is at the heart of many modern technologies, including navigation, (the global positioning system), electric power generation (synchronization of generators feeding into national power grids), and telecommunication (synchronous data transfers, financial transactions). Scientifically, clock synchronization is key to projects such as long baseline interferometry (distributed radio telescopes), gravitational wave observation (LIGO), tests of the general theory of relativity, and distributed computation.The basic problem is easily formulated: determine the time difference ∆ between two spatially separated clocks, using the minimum communication resources. Generally, the accuracy to which ∆ can be determined is a function of the clock frequency stability, and the uncertainty in the delivery times for messages sent between the two clocks. Given the stability of present clocks, and assuming realistic bounded uncertainties in the delivery times (e.g. satellite to ground transmission delays), protocols have been developed which presently allow determination of ∆ to accuracies better than 100 ns (even for clock separations greater than 8000 km); it is also predicted that accuracies of 100 ps should be achievable in the near future.However, these protocols fail if the message delivery time is too uncertain, because they rely upon the law of large numbers to achieve a constant average delivery time (thus, also requiring O(2 2n ) messages to obtain n digits of ∆). If the required averaging time is longer than the stability time of the local clocks, then these protocols must be replaced. A simple, different, protocol, which succeeds independent of the delivery time, is to just send a clock which keeps track of the delivery time. For example, if Alice mails Bob a wristwatch synchronized to her clock, then when Bob receives it he can clearly calculate the ∆ for their two clocks from the difference between his time and that given by the wristwatch. This wristwatch protocol is generally impractical, but it suggests another scheme which is intriguing. A quantum bit (qubit) behaves naturally much like a small clock. For example, a nuclear spin in a magnetic field precesses at a frequency given by its gyromagnetic ratio times the magnetic field strength. And an optical qubit, represented by the the presence or absence of a single photon in a given mode, oscillates at the frequency of the electromagnetic carrier. The relative phase between the |0 and |1 states of a qubit thus keeps time, much like a clock, and ticks ...