  The Quantum Satisfiability problem (QSAT) is the generalization of the
canonical NP-complete problem - Boolean Satisfiability. (k,s)-QSAT is the
following variant of the problem: given a set of projectors of rank 1, acting
non-trivially on k qubits out of n qubits, such that each qubit appears in at
most s projectors, decide whether there exists a quantum state in the null
space of all the projectors. Let f*(k) be the maximal integer s such that every
(k,s)-QSAT instance is satisfiable. Deciding (k,f*(k))-QSAT is computationally
easy: by definition the answer is "satisfiable". But, by relaxing the
conditions slightly, we show that (k,f*(k)+2)-QSAT is QMA_1-hard, for k >=15.
This is a quantum analogue of a classical result by Kratochv\'il et al.
[KST93]. We use the term "an almost sudden jump" to stress that the complexity
of (k,f*(k)+1)-QSAT is open, where the jump in the classical complexity is
known to be sudden.
  We present an implication of this finding to the quantum PCP conjecture,
arguably one of the most important open problems in the field of Hamiltonian
complexity. Our implications impose constraints on one possible way to refute
the quantum PCP.
