Summary: We introduce an \(n\)-species totally asymmetric zero range process (\(n\)-TAZRP) on one-dimensional periodic lattice with \(L\) sites. It is a continuous time Markov process in which \(n\) species of particles hop to the adjacent site only in one direction under the condition that smaller species ones have the priority to do so. Also introduced is an \(n\)-line process, a companion stochastic system having the uniform steady-state from which the \(n\)-TAZRP is derived as the image by a certain projection \(\pi\). We construct the \(\pi\) by a combinatorial \(R\) of the quantum affine algebra \(U_q(\widehat{sl}_L)\) and establish a matrix product formula of the steady state probability of the \(n\)-TAZRP in terms of corner transfer matrices of a \(q=0\)-oscillator-valued vertex model. These results parallel the recent reformulation of the \(n\)-species totally asymmetric simple exclusion process (\(n\)-TASEP) by the authors, demonstrating that \(n\)-TAZRP and \(n\)-TASEP are the canonical sister models associated with the symmetric and the antisymmetric tensor representations of \(U_q(\widehat{sl}_L)\) at \(q=0\), respectively.