On Inverted Earth, an experience is caused by red things, used in thought about red things, and guides action with respect to red things — satisfying the functionalist's necessary and sufficient conditions for possessing Q_R.
functionalism(Philosophy of mind; distinguished here from representationalism)
The view that mental states are defined by the causal roles they play in a cognitive system — their actual, potential, or typical causal relationships.
necessary and sufficient conditions(in philosophical analysis)
A 'necessary' condition is something that must be true for something else to happen; a 'sufficient' condition is something that guarantees it will happen. This phrase describes what must be true (and what's enough) for a definition to apply.
This gives us an apparent counterexample against the “long arm” functionalist theory. On Inverted Earth your experiences have the functional role that the functionalist theory says is necessary and sufficient for possessing \(Q_{R}\): they are “caused by red things in the right circumstances, and used in thought about red things and action with respect to red things rightly”. But on Inverted Earth your experiences with this functional role have \(Q_{G}\), not (or not always) \(Q_{R}\).