We introduce a (0.611)-approximation algorithm for Quantum MaxCut and a (\frac{1+\sqrt{5}}{4} \approx 0.809)-approximation algorithm for the EPR Hamiltonian of [arXiv:2209.02589]. A novel ingredient in both of these algorithms is to partially entangle pairs of qubits associated to edges in a matching, while preserving the direction of their single-qubit Bloch vectors. This allows us to interpolate between product states and matching-based states with a tunable parameter.