I will show the video Quantum levitation, and discuss what is and isn't quantum about it.
A good discussion of some of the issues raised is Laughlin and Pines article The Theory of Everything. A more extensive and introductory discussion by Pines is at Physics for the 21st Century.
Postscript.
Based on comments and questions afterwards, particularly from some undergraduates, there are few things I would do slightly differently.
I should have said what a Hamiltonian is: a function that defines the energy as a function of the system variables, e.g., the position and velocity of all of the particles.
The stratification of reality shown by my boxes is a simplification for schematic purposes. There is no clearly defined boundary between strata. For example, at the boundary between chemistry and physics one has chemical physics and physical chemistry. The boundary between biology and biochemistry is blurred. On the other hand, anatomy is qualitatively different to enzyme mechanisms. Acid-base equilibria is chemistry not physics.
Ben Powell emphasized to me that the claim that "superconductors exhibit broken U(1) gauge symmetry" is problematic and subtle. There is a long detailed paper, Superconductors are topologically ordered that I have read several times but don't really understand.
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