Identifying properties in quantum dynamics, such as identity or time-reversal invariance, is a crucial challenge with profound implications for quantum technologies. We present a unified framework that integrates group representation theory with quantum hypothesis testing to optimally predict the fundamental properties of quantum dynamics. By exploiting the inherent symmetry of compact groups and their irreducible representations, we derive an exact characterization of the optimal type-II error (failure probability to detect a symmetry), offering an operational interpretation for the quantum maximum-relative entropy. In particular, we prove that parallel strategies achieve the same performance as adaptive or indefinite-causal-order protocols, resolving debates about the necessity of complex control sequences. Examples to the singleton group, maximal commutative group, and orthogonal group yield explicit results: for predicting the identity, Z-symmetry, and T-symmetry properties of unknown qubit unitaries, with zero type-I error and type-II error bounded by $\delta$, we establish the explicit optimal sample complexity which scales as $\mathcal{O}\left({\delta }^{-1/3}\right)$ for identity testing and $\mathcal{O}\left({\delta }^{-1/2}\right)$ for $T/Z$-symmetry testing. These findings offer theoretical insights and practical guidelines for efficient unitary property testing and symmetry-driven protocols in quantum information processing.

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• Quantum algorithms
• Quantum correlations in quantum information
• Quantum information theory
• Quantum protocols
• Quantum verification