The development of precise atomic clocks has led to many scientific and technological advances that play an increasingly important role in modern society. Shared timing information constitutes a key resource for positioning and navigation with a direct correspondence between timing accuracy and precision in applications such as the Global Positioning System (GPS). By combining precision metrology and quantum networks, we proposed a quantum, cooperative protocol for the operation of a network consisting of geographically remote optical atomic clocks. Through use of non-local entangled states, we demonstrated an optimal utilization of the global network resources, and showed that such a network could be operated near the fundamental limit set by quantum theory that yielded an ultra-precise clock signal. Furthermore, the internal structure of the network, combined with basic techniques from quantum communication, guaranteed security both from internal and external threats. Realization of such a global quantum network of clocks may allow construction of a real-time single international time scale (world clock) with unprecedented stability and accuracy.