Apparently the quantum effect for the later (3) and (4) becomes important. But what physical phenomenon of the (1), (2), (3), (4) set their differences?
Spin ice: Possible candidates include that Dy$_2$ Ti$_2$ O$_7$ and Ho$_2$Ti$_2$O$_7$, etc. may harbor a classical spin ice. They potentially satisfy an ‘ice rule’: each tetrahedron of the pyrochlore lattice are precisely two spins that point inward and two that point outward. It may also have magnetic monopole if we have 3-in 1-out tetrahedron or anti-monopole for 1-in 3-out tetrahedron. In classical spin ice systems may be thermally fluctuating loop gas. The loops can be ‘magnetic’ field lines of an artificial magnetic fields.
Quantum spin ice: Possible candidates include Tb$_2$Ti$_2$O$_7$, Yb$_2$Ti$_2$O$_7$, Pr$_2$Zr$_2$O$_7$, etc. In quantum spin ice, the physics is determined by quantum fluctuations of oriented loops. If these loops form a liquid phase where the loop line tension becomes zero, then it becomes a quantum spin liquid. But how this spin liquid supports an emergent gapless photon? The associated magnetic field lines can be tensionless. Magnetic monopoles as the defect (in tetrahedra) can be gapped quasiparticle excitations where these field lines end. But how the gapless photons can emerge from the quantum fluctuation from spin ice?
Quantum spin liquids: Possible candidates include herbertsmithite.
How can quantum spin liquid have emergent photons, but not for spin ice or quantum spin ice?
For example, can one understand from here: If the loops form a spin ice's liquid phase where the loop line tension becomes zero, then it becomes a quantum spin liquid. And how the photon emerges from here?