Post-Quantum Extension of the Conditional CRO Trilemma: Impossibility Bounds on Confidentiality, Reliability, and Opposability Under Quantum Institutional Extraction Limits

Abstract

Cryptographic evidence systems face fundamental trade-offs between confidentiality (Conf), reliability (Rel), and opposability (Opp), with Conf $\cdot $ Rel $\cdot $ Opp products ranging from 0.27 to 0.57 in deployed systems. The Conditional CRO Trilemma recently established that institutional capacity $\Delta $ fundamentally constrains achievable trade-offs through the bound $\text {Conf}\cdot \text {Rel}\cdot \text {Opp} \leq \Delta /\lambda + \mathcal {O}(\lambda ^{-2})$ . However, the imminent deployment of post-quantum cryptographic systems raises critical questions: do quantum adversaries and quantum channel effects modify these bounds? This paper presents the Post-Quantum Conditional CRO Trilemma, establishing that both institutional capacity and quantum interpretability loss fundamentally constrain achievable trade-offs. Under explicit assumptions about quantum-aware institutional limits, we prove \begin{equation*} \text {Conf}^{q} \cdot \text {Rel}^{q} \cdot \text {Opp}^{q} \leq \frac {\Delta _{q} + \eta _{q} \cdot H(C)}{\lambda } + \mathcal {O}(\lambda ^{-2}) + \text {negl}(\lambda)\end{equation*} where $\Delta _{q}$ quantifies the quantum-aware institutional extraction capacity, $\eta _{q}$ is the quantum interpretability loss coefficient, $H(C)$ is the contextual entropy, and $\lambda $ is the security parameter. Our main contributions are as follows: 1) a quantum extension of the institutional capacity model with parameters $\Delta _{q}$ and $\eta _{q}$ ; 2) complete information-theoretic proofs using Quantum Pinsker’s inequality, the Gentle Measurement Lemma, and quantum Fano bounds; 3) an internal coherence analysis showing that NIST PQC finalists (Dilithium, Kyber, and SPHINCS+) imply $\Delta _{q} \in [{28, 67}]$ bits under quantum adversaries, consistent with quantum-aware bounded rationality; and 4) a framework for matching post-quantum protocols to quantum-aware institutional capacity. We demonstrate the internal coherence of our framework against the same deployed systems analyzed in the Conditional CRO work, Helios voting, zk-SNARKs compliance, and ECDSA signatures, showing how quantum effects modify their implied institutional capacities. This trilemma shifts post-quantum protocol design from purely computational security to quantum-constraint-aware engineering, with applications to GDPR-compliant quantum-safe systems and eIDAS quantum readiness.