Using the discrete zero‐curvature equation, we derive a hierarchy of new nonlinear differential–difference equations associated with a discrete 3×3$$ 3\times 3 $$ matrix spectral problem. Resorting to the characteristic polynomial of Lax matrix, we introduce a trigonal curve and the associated three‐sheeted Riemann surface, from which we derive the Baker–Akhiezer function and meromorphic function. By comparing the asymptotic expansions of the meromorphic function and its Riemann theta function representations, we obtain quasi‐periodic solutions for a hierarchy of nonlinear differential–difference equations.