We investigate magnon-magnon entanglement and synchronization in a cavity-magnon system comprising two yttrium iron garnet spheres, focusing on the role of symmetric and asymmetric magnon pairs under varying physical parameters, including cavity detuning, photon-magnon coupling, dissipation rates, and thermal noise. Our results demonstrate that symmetric magnon pairs generally favor the generation of both quantum entanglement and synchronization, with enhanced resilience to thermal fluctuations compared to asymmetric configurations. Particularly, the quantum entanglement and synchronization obtained in both scenarios (symmetric and asymmetric magnon pairs) can exhibit vastly different behaviors with changes in some physical factors. Further, we explore the interplay between magnon-magnon quantum entanglement and synchronization. We identify a competitive relationship between entanglement and synchronization in certain situations, manifesting as parameter-dependent complementarity where enhancement of one effect correlates with suppression of the other. This study underscores the critical influence of symmetry configuration on quantum correlations in hybrid cavity-magnon systems, offering theoretical guidance for optimizing entanglement and synchronization in quantum information platforms.

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• Open quantum systems & decoherence
• Quantum cavities
• Quantum coherence & coherence measures
• Quantum correlations in quantum information
• Quantum correlations, foundations & formalism