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Information entropies give a genuine way to characterize quantitatively anincompatibility in quantum measurements. Together with the Shannon entropy, fewfamilies of parametrized entropies have found use in various questions. It isalso known that a possibility to vary the parameter can often provide morerestrictions on elements of probability distributions. In quantum informationprocessing, one often deals with measurements having some special structure.Quantum designs are currently the subject of active research, whence the aim toformulate complementarity relations for related measurements occurs. Usinggeneralized entropies of order $\alpha\in(0,2)$, we obtain uncertainty andcertainty relations for POVMs assigned to a quantum design. The structure ofquantum designs leads to several restrictions on generated probabilities. Weshow how to convert these restrictions into two-sided entropic estimates. Oneof the used ways is based on truncated expansions of the Taylor type. Therecently found method to get two-sided entropic estimates uses polynomials withflexible coefficients. We illustrate the utility of this method with respect toboth the R\'{e}nyi and Tsallis entropies. Possible applications of the derivedcomplementarity relations are briefly discussed.

Complementarity relations for design-structured POVMs in terms of generalized entropies of order $\alpha\in(0,2)$