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This paper presents a novel method for computing certain particle locationsin the multi-species $q$-TAZRP (totally asymmetric zero range process). Themethod is based on a decomposition of the process into its discrete-timeembedded Markov chain, which is described more generally as a monotone processon a graded partially ordered set; and an independent family of exponentialrandom variables. A further ingredient is explicit contour integral formulasfor the transition probabilities of the $q$-TAZRP. The main result of thismethod is a shift invariance for the multi-species $q$-TAZRP on the infiniteline.  By a previously known Markov duality result, these particle locations are thesame as joint $q$-moments. One particular special case is that for step initialconditions, ordered multi-point joint $q$-moments of the $n$-species $q$-TAZRPmatch the $n$-point joint $q$-moments of the single-species $q$-TAZRP. Thus, weconjecture that the Airy$_2$ process describes the joint multi-pointfluctuations of multi-species $q$-TAZRP.  As a probabilistic application of this result, we find explicit contourintegral formulas for the joint $q$-moments of the multi-species $q$-TAZRP inthe diffusive scaling regime.

Joint $q$-moments and shift invariance for the multi-species $q$-TAZRP on the infinite line