QMA= QMA₁ with an infinite counter

A long-standing open problem in quantum complexity theory is whether QMA, the quantum analogue of NP, is equal to QMA₁, its one-sided error variant. We show that QMA= QMA∞= QMA₁∞, where QMA₁^∞ is like QMA₁, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an ``infinite counter'', and compare it to a program counter in a Las Vegas algorithm. The result QMA= QMA^∞ means such an infinite register does not increase the power of QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous QMA amplifiers. Our new construction achieves completeness 1-2-q using $O(1)$ calls to each of the original verifier and its inverse, and O(log q) other gates, proving that QMA has completeness doubly exponentially close to 1, i.e. QMA= QMA(1-2-2ʳ,2⁻ʳ) for any polynomial r.