Quantum PCP and hardness of simulating of Hamiltonians
As Shor said, there is no QPCP theorem (yet!).  One conjecture (let us call it the QPCP conjecture) is this: consider a graph of N vertices, of degree O(1).  Associate a qudit with each vertex, with Hilbert space dimension O(1).  Let the Hamiltonian be a sum of terms for each edge, each such term acting just on the qudits on the vertices, with operator norm of each such term bounded by O(1), so that the operator norm of the Hamiltonian is O(N).  Then, the conjecture is that there is some epsilon>0 such that it is QMA-hard to approximate the ground state energy of the problem to an accuracy epsilon N.

A slightly stronger conjecture is to consider the case in which each such term acting on an edge is a projector so that the ground state energy is non-negative, and the conjecture is that it is QMA-hard to determine whether the ground state energy is 0 given a promise that if it is not zero then it is at least epsilon N.

There are other versions of the conjecture too, but those are two interesting ones with the most natural relation to physics.  An even stronger conjecture (hence probably an easier one to disprove if you believe that these conjectures are false) is to consider the case in which the Hamiltonian is a sum of commuting projectors.