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Quantum graphs are an operator space generalization of classical graphs thathave emerged in different branches of mathematics including operator theory,non-commutative topology and quantum information theory. In this paper, weobtain lower bounds for the classical and quantum chromatic number of a quantumgraph using eigenvalues of the quantum adjacency matrix. In particular, weprove a quantum generalization of Hoffman's bound and introduce quantumanalogues for the edge number, Laplacian and signless Laplacian. We generalizeall the spectral bounds given by Elphick and Wocjan (2019) to the quantum graphsetting and demonstrate the tightness of these bounds in the case of completequantum graphs. Our results are achieved using techniques from linear algebraand a combinatorial definition of quantum graph coloring, which is obtainedfrom the winning strategies of a quantum-to-classical nonlocal graph coloringgame, given by M. Brannan, P. Ganesan and S. Harris (2020).

Spectral bounds for the quantum chromatic number of quantum graphs