Despite recent criticism, the search for neural correlates of consciousness (NCCs) is still at the core of a contemporary neuroscience of consciousness. One common aim is to distinguish merely statistical correlates from “NCCs proper”, i.e., NCCs that are uniquely associated with a conscious experience and lend themselves to a metaphysical interpretation. We should then distinguish between NCCs as data and NCCs as hypotheses, where the first is just recorded data while the second goes beyond any set of recorded data. Still, such NCC-hypotheses ought to be testable. Here, I present a framework for so-called “sufficiency tests.” We can distinguish four different classes of such tests, depending on whether they predict creature consciousness (which systems are conscious), state consciousness (when a system is conscious), phenomenal content (what a system is conscious of), or phenomenal character (how a system experiences). For each kind of test, I provide examples from the empirical literature. I also argue that tests for phenomenal character (How-Tests) are preferable because they bracket problematic aspects of the other kinds of tests. However, How-Tests imply a metaphysical tie between the neural and phenomenal domain that is stronger than supervenience, delivers explanations but does not close the explanatory gap, uses first-person methods to test hypotheses, and thereby relies on a form of direct neurophenomenal structuralism.