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The proliferation of mobile networks and their increasing importance to modern life, combined with the emerging threat of quantum computing, present new challenges and opportunities for cybersecurity. This paper addresses the complexity of protecting these critical infrastructures against future quantum attacks while considering operational sustainability. We begin with an overview of the current landscape, identify the main vulnerabilities in mobile networks, and evaluate existing security solutions with new post-quantum cryptography (PQC) methods. We then present a quantum-secure architecture with PQC and Quantum Key Distribution (QKD) tailored explicitly for sustainable mobile networks and illustrate its applicability with several use cases that emphasize the need for advanced protection measures in this new era. In addition, a comprehensive analysis of PQC algorithm families is presented, focusing on their suitability for integration in mobile environments, with particular attention to the trade-offs between energy consumption and security improvements. Finally, recommendations for strengthening mobile networks against quantum threats are provided through a detailed examination of current challenges and opportunities.