  The concept of quantum coherence and its possible use as a resource are
currently the subject of active researches. Uncertainty and complementarity
relations for quantum coherence allow one to study its changes with respect to
other characteristics of the process of interest. Protocols of quantum
information processing often use measurements that have a special inner
structure. Quantum designs are considered as tools with a perspective of
fruitful applications in quantum information processing. We obtain uncertainty
and certainty relations for coherence averaged with respect to a set of
design-structured POVMs of rank one. To characterize the degree of coherence,
the relative entropy of coherence is utilized. The derived relations are
exemplified with quantum designs in the two-dimensional Hilbert space.
