The precise synchronization of distant clocks is a fundamental requirement for a wide range of applications. Here, we experimentally demonstrate an approach of quantum clock synchronization by distributing entangled and correlated photon pairs from a telecom-wavelength quantum dot over a metropolitan fiber network in the Stockholm area. By leveraging the tight time correlation between the emitted photons, we achieve a synchronization accuracy of tens of picoseconds. We show that our synchronization scheme is secure against spoofing attacks by performing a remote quantum state tomography to verify the origin of the entangled photons. We measured a distributed maximum entanglement fidelity of 0.817 +/- 0.040 to the |Phi+> Bell state and a concurrence of 0.660 +/- 0.086. These results highlight the potential of quantum dot-generated entangled pairs as a shared resource for secure time synchronization and quantum key distribution in real-world quantum networks. Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.