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We formulate the pentagon relation for quantum dilogarithm elements in thestructure group of a quantum cluster scattering diagram (QCSD). As anapplication, we show the nonpositivity of a certain class of nonskew-symmetricQCSDs. Also, we explicitly present various consistency relations for QCSDs ofrank 2 completely or up to some degree, many of which are new in theliterature.

Pentagon relation in quantum cluster scattering diagrams