Abstract: At CRYPTO 2013, Boneh and Zhandry initiated the study of quantum-secure encryption. They proposed first indistinguishability definitions for the quantum world where the actual indistinguishability only holds for classical messages, and they provide arguments why it might be hard to achieve a stronger notion. In this work, we show that stronger notions are achievable, where the indistinguishability holds for quantum superpositions of messages. We investigate exhaustively the possibilities and subtle differences in defining such a quantum indistinguishability notion. We justify our stronger definition by showing their equivalence to novel quantum semantic-security notions that we introduce. Furthermore, we give a generic transformation to turn a big class of encryption schemes into quantum indistinguishable and hence quantum semantically secure ones.