We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum homomorphisms and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non-commutative graphs with respect to entanglement-assisted homomorphisms. We apply Strassen's spectral theorem (J. Reine Angew. Math., 1988) and obtain dual characterizations of the corresponding Shannon capacities and asymptotic preorders in terms of their asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam (Combinatorica, 2019) to the quantum domain. We study the relations among the three new quantum asymptotic spectra and the asymptotic spectrum of graphs. The bounds on the several Shannon capacities that have appeared in the literature we fit into the corresponding quantum asymptotic spectra. In particular, we prove that the (fractional) complex Haemers bound upper bounds the quantum Shannon capacity, defined as the regularization of the quantum independence number (Mančinska and Roberson, J. Combin. Theory Ser. B, 2016), which leads to a separation with the Lovász theta function.