Interpretation of the curious results of the new quantum formalism of pre- and post-selected systems

The analysis, with the use of two state vectors, of a quantum system, during the time interval between two measurements, leads to some amazing results, which seem to contradict our usual quantum common sense. We explore the questions of compatibility with the conventional quantum theory, uniqueness of pre- and post-selected ensembles, commutativity, simultaneity and reality of strong and weak values in the intermediate time, and the meaning of the weak value. Common criticisms are shown to be unfounded.