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We investigate the additivity properties for both bipartite and multipartitesystems by using entropic uncertainty relations (EUR) defined in terms of thejoint Shannon entropy of probabilities of local measurement outcomes and weapply them to entanglement detection. In particular, we introducestate-independent and state-dependent entropic inequalities whose violationcertifies the presence of quantum correlations. We show that the additivity ofEUR holds only for EUR that involve two observables, while inequalities thatconsider more than two observables or the addition of the Von Neumann entropyof a subsystem enable to detect quantum correlations. Furthermore, we studytheir detection power for bipartite systems and for several classes of statesof a three-qubit system.

Tight bounds from multiple observables entropic uncertainty relations