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Quarkonium bound states are especially promising candidates to test the probable quantum structure of space-time, since they represent a system with reasonably small characteristic distance. Although the theory of quarkonia is the quantum chromodynamics, the interaction between the quarks can be heuristically described by the quantum mechanical Cornell potential as well. Firstly, we investigate the quarkonium system in the 3-dimensional smooth configuration space. Thereafter, we insert it into a non-commutative 3-dimensional configuration space which is composed of concentric fuzzy spheres of increasing radius, called the fuzzy onion, in order to extract some consequences of the non-trivial structure of the space by means of the quarks' properties.