The spectrum of a commutative algebra is a geometric or topological space on which the algebra may be viewed as a ring of functions. Spectrum constructions for various classes of commutative algebras famously provide a bridge between algebra and geometry, serving as one of the primary inspirations for various flavors of noncommutative geometry. But what kind of object should fill the role of the spectrum of a noncommutative algebra? In recent years, a number of results have ruled out both naive and subtle attempts to resolve this problem. This suggests that we frame the problem in light of a more fundamental question: What objects should serve as quantum discrete spaces in noncommutative geometry? In this talk, I will survey various obstructions and partial progress on these problems in both ring theory and operator algebra.