  We formulate some properties of a set of several mutually unbiased
measurements. These properties are used for deriving entropic uncertainty
relations. Applications of mutually unbiased measurements in entanglement
detection are also revisited. First, we estimate from above the sum of the
indices of coincidence for several mutually unbiased measurements. Further, we
derive entropic uncertainty relations in terms of the R\'{e}nyi and Tsallis
entropies. Both the state-dependent and state-independent formulations are
obtained. Using the two sets of local mutually unbiased measurements, a method
of entanglement detection in bipartite finite-dimensional systems may be
realized. A certain trade-off between a sensitivity of the scheme and its
experimental complexity is discussed.
