  We consider a model of quantum computation using qubits where it is possible
to measure whether a given pair are in a singlet (total spin $0$) or triplet
(total spin $1$) state. The physical motivation is that we can do these
measurements in a way that is protected against revealing other information so
long as all terms in the Hamiltonian are $SU(2)$-invariant. We conjecture that
this model is equivalent to BQP. Towards this goal, we show: (1) this model is
capable of universal quantum computation with polylogarithmic overhead if it is
supplemented by single qubit $X$ and $Z$ gates. (2) Without any additional
gates, it is at least as powerful as the weak model of "permutational quantum
computation" of Jordan [14, 18]. (3) With postselection, the model is
equivalent to PostBQP.
