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This paper introduces a novel lower bound on communication complexity using quantum relative entropy and mutual information, refining previous classical entropy-based results. By leveraging Uhlmann's lemma and quantum Pinsker inequalities, the authors establish tighter bounds for information-theoretic security, demonstrating that quantum protocols inherently outperform classical counterparts in balancing privacy and efficiency. Also explores symmetric Quantum Private Information Retrieval (QPIR) protocols that achieve sub-linear communication complexity while ensuring robustness against specious adversaries: A post-quantum cryptography based protocol that can be authenticated for the specious server; A ring-LWE-based protocol for post-quantum security in a single-server setting, ensuring robustness against quantum attacks; A multi-server protocol optimized for hardware practicality, reducing implementation overhead while maintaining sub-linear efficiency. These protocols address critical gaps in secure database queries, offering exponential communication improvements over classical linear-complexity methods. The work also analyzes security trade-offs under quantum specious adversaries, providing theoretical guarantees for privacy and correctness.