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The main aim of this article is to characterize inner Poisson structure on aquantum cluster algebra without coefficients. Mainly, we prove that innerPoisson structure on a quantum cluster algebra without coefficients is always astandard Poisson structure. In order to relate with compatible Poissonstructure, we introduce the concept of so-called locally inner Poissonstructure on a quantum cluster algebra and then show it is equivalent tolocally standard Poisson structure in the case without coefficients. Based onthe result from \cite{LP} we obtain finally the equivalence between locallyinner Poisson structure and compatible Poisson structure in this case.

On inner Poisson structures of a quantum cluster algebra without coefficients