  The uncertainty principle restricts potential information one gains about
physical properties of the measured particle. However, if the particle is
prepared in entanglement with a quantum memory, the corresponding entropic
uncertainty relation will vary. Based on the knowledge of correlations between
the measured particle and quantum memory, we have investigated the entropic
uncertainty relations for two and multiple measurements, and generalized the
lower bounds on the sum of Shannon entropies without quantum side information
to those that allow quantum memory. In particular, we have obtained
generalization of Kaniewski-Tomamichel-Wehner's bound for effective measures
and majorization bounds for noneffective measures to allow quantum side
information. Furthermore, we have derived several strong bounds for the
entropic uncertainty relations in the presence of quantum memory for two and
multiple measurements. Finally, potential applications of our results to
entanglement witnesses are discussed via the entropic uncertainty relation in
the absence of quantum memory.
