Quantum key distribution (QKD) security fundamentally relies on the ability to distinguish genuine quantum correlations from classical eavesdropper simulations, yet existing certification methods lack rigorous statistical guarantees under finite-sample conditions and adversarial scenarios. We introduce TARA (Test by Adaptive Ranks), a novel framework combining conformal prediction with sequential martingale testing for quantum anomaly detection that provides distribution-free validity guarantees. TARA offers two complementary approaches. TARA k, based on Kolmogorov Smirnov calibration against local hidden variable (LHV) null distributions, achieving ROC AUC = 0.96 for quantum-classical discrimination. And TARA-m, employing betting martingales for streaming detection with anytime valid type I error control that enables real time monitoring of quantum channels. We establish theoretical guarantees proving that under (context conditional) exchangeability, conformal p-values remain uniformly distributed even for strongly contextual quantum data, confirming that quantum contextuality does not break conformal prediction validity a result with implications beyond quantum certification to any application of distribution-free methods to nonclassical data. Extensive validation on both IBM Torino (superconducting, CHSH = 2.725) and IonQ Forte Enterprise (trapped ion, CHSH = 2.716) quantum processors demonstrates cross-platform robustness, achieving 36% security margins above the classical CHSH bound of 2. Critically, our framework reveals a methodological concern affecting quantum certification more broadly: same-distribution calibration can inflate detection performance by up to 44 percentage points compared to proper cross-distribution calibration, suggesting that prior quantum certification studies using standard train test splits may have systematically overestimated adversarial robustness.