The efficiency of Shor's and Grover's algorithms and the advancement of quantum computers implies that the cryptography used until now to protect one's privacy is potentially vulnerable to retrospective decryption, also known as the harvest now, decrypt later attack in the near future. This dissertation proposes an overview of the cryptographic schemes used by Tor, highlighting the non-quantum-resistant ones and introducing theoretical performance assessment methods of a local Tor network. The measurement is divided into three phases. We start with benchmarking a local Tor network simulation on constrained devices to isolate the time taken by classical cryptography processes. Secondly, the analysis incorporates existing benchmarks of quantum-secure algorithms and compares these performances on the devices. Lastly, the estimation of overhead is calculated by replacing the measured times of traditional cryptography with the times recorded for Post-Quantum Cryptography (PQC) execution within the specified Tor environment. By focusing on the replaceable cryptographic components, using theoretical estimations, and leveraging existing benchmarks, valuable insights into the potential impact of PQC can be obtained without needing to implement it fully.