Homomorphic encryption (HE) schemes enable the processing of encrypted data and may be used by a user to outsource storage and computations to an untrusted server. A plethora of HE schemes has been suggested in the past four decades, based on various assumptions, which achieve different attributes. In this work, we assume that the user and server are quantum computers and look for HE schemes of classical data. We set a high bar of requirements and ask what can be achieved under these requirements. Namely, we look for HE schemes which are efficient, information-theoretically secure, perfectly correct, and which support homomorphic operations in a fully compact and non-interactive way. Fully compact means that decryption costs O(1 ) time and space. We suggest an encryption scheme based on random bases and discuss the homomorphic properties of that scheme. The main advantage of our scheme is providing better security in the face of weak measurements (WM). Measurements of this kind enable collecting partial information on a quantum state while only slightly disturbing the state. We suggest here a novel QKD scheme based on our encryption scheme, which is resilient against WM-based attacks. We bring up a new concept we call securing entanglement. We look at entangled systems of qubits as a resource used for carrying out quantum computations and show how our scheme may be used to guarantee that an entangled system can be used only by its rightful owners. To the best of our knowledge, this concept has not been discussed in previous literature.