We argue that the photon spectra in radiative decays of various heavy quarkonium states provide important information on their nature. If two of these states are in the strong coupling regime, we are able to produce a parameter-free model-independent formula, which holds at next-to-leading order and includes both direct and fragmentation contributions. When the formula is checked against recent CLEO data it favors 2S and 3S in the strong coupling regime and disfavors 1S in it. DOI: 10.1103/PhysRevLett.96.111801 PACS numbers: 13.20.Gd, 12.39.St Heavy quarkonium systems have played a major role in our understanding of QCD (see [1] for a review). Inclusive heavy quarkonium decays to light particles involve a short distance process, in which the heavy quark and antiquark annihilate into gluons or photons, and a long distance process, which takes into account that the heavy quark and antiquark are bound in a color singlet state. Because of the asymptotic freedom of QCD the short distance process can be calculated using perturbation theory in s m , m being the heavy quark mass. It was believed for some time that in order to have a color singlet state the heavy quark and antiquark in it had to be in a color singlet state themselves. This allowed us to parameterize the long distance process by a wave function at the origin (or derivatives of it). In spite of the fact that this wave function was not computable from perturbative QCD (pQCD), it canceled in suitable ratios, and hence predictions depending on pQCD only could be put forward. Nevertheless, it was soon noticed that the above framework had difficulties due to the infrared divergences which showed up at higher orders of s m [2]. The introduction of color octet operators in the framework of nonrelativistic QCD (NRQCD) [3] provided a rigorous way to understand and to deal with these divergences [4]. The price to be paid was the introduction of new nonperturbative parameters, the color octet matrix elements, which made predictions depending on pQCD only more difficult to obtain.Later on, it was pointed out that of the hierarchy of relevant scales in heavy quarkonium m mv mv 2 , v being the relative velocity, only the first inequality was exploited in NRQCD and that the velocity counting proposed in [3] did not necessarily follow from it. It was also shown how the last inequality could be exploited by constructing a further effective theory, namely, potential NRQCD (pNRQCD) [5,6](see [7] for a review). The interplay of QCD with the scales mv and mv 2 above dictates the degrees of freedom of pNRQCD. Two regimes have been identified: (i) the weak coupling regime, QCD & mv 2 , and (ii) the strong coupling regime mv 2 QCD & mv. In the weak coupling regime, the relevant degrees of freedom are a singlet and an octet wave function fields interacting with (perturbative) potentials and with ultrasoft gluons. The original NRQCD counting [3] belongs to this regime. Moreover, the use of diagrammatic and quantum mechanical methods allows us to carry out explicit calculations ...