We investigate the variability of the quantum coherences of an arbitrary density matrix, in a given basis. An easily computable quantity that captures this variability notion is proposed and a dual relationship with an appropriate entropy is observed, with coherence dispersion (CD) being maximized for intermediate values of disorder and vanishing for extremal entropy values. This is a prevalent signature of complexity quantifiers across different fields, from evolutionary biology to information science. In addition, the physical significance of CD is investigated in the context of entropy production in open systems. We also study a specific family of nonequilibrium systems for which coherence dispersion presents a peaked maximum for finite temperatures. The final part of the manuscript is dedicated to the application of CD as a key ingredient to establish quite general quantum speed limits in the Hilbert space.

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• Open quantum systems & decoherence
• Quantum foundations