In the context of 6G architecture development, the concept of a softwarized (orchestration) continuum is a key pillar. Nevertheless, achieving complete softwarization of network functionalities, tasks, and operations presents inherent challenges, leading to critical trade-offs and limitations. This article explores a novel approach to address these issues by integrating quantum technologies and the Physical Layer Service Integration (PLSI) paradigm. Specifically, we propose the formulation and analysis of network synchronization as a quantum PLSI problem. Our study evaluates synchronization time offset in both conventional Precision Time Protocol (PTP) and quantum-based approaches within the network. We investigate the impact of various network conditions on the precision of PTP synchronization, ranging from nanoseconds under ideal circumstances to microseconds when utilizing virtual network devices. Further, we perform a simulation to generate frequency-entangled photon pairs to access nonlocal temporal correlations and calculate the time offsets. Our findings reveal that entanglement-based PLSI for network synchronization achieves precision at the picosecond level. These results emphasises the high precision achievable by interpreting the network synchronisation problem in the perspective of PLSI and not as a service of the softwarized continuum.