In order to maintain a similar security level in a post-quantum setting, many symmetric primitives should have to double their keys and increase their state sizes. So far, no generic way for doing this is known that would provide convincing quantum security guarantees. In this paper we propose a new generic construction, QuEME, that allows to double the key and the state size of a block cipher. The QuEME design is inspired by the ECB-Mix-ECB (EME) construction, but is defined for a different choice of mixing function that withstands our new quantum superposition attack that exhibits a periodic property found in collisions and that breaks EME and a large class of variants of it. We prove that QuEME achieves n-bit security in the classical setting, where n is the block size of the underlying block cipher, and at least n/6-bit security in the quantum setting. We propose a concrete instantiation of this construction, called Double-AES, that is built with variants of AES-128.