This is proven by using techniques from complex approximation theory, to make the oracle separation from (Aaronson, 2008), between QMA and QMA with perfect completeness, quantitative. To explain the context: QMA, or Quantum Merlin Arthur, is the canonical quantum version of NP. In other words, does QMA = QMA 1 , where QMA 1 is the subclass of QMA that admits protocols with “perfect completeness”? In 2008, I used real analysis to show that there’s a quantum oracle relative to which QMA ≠ QMA 1 , which means that any proof of QMA = QMA 1 would need to use “quantumly nonrelativizing techniques” (not at all an insuperable barrier, but at least we learned something about why the problem is nontrivial). In other words: we showed that, when one makes my 2008 QMA ≠ QMA 1 quantum oracle separation quantitative, one gets a lower bound that precisely matches Freek and Stacey’s protocol.