> The catch is that appearance of classical objects is still unsolved problem in QM.
What? Decoherence explains classical objects.
What? Decoherence explains classical objects.
1, which is very practical, is that it doesn't explain why the wave function always decoheres in the same way. In experiments, you can always chose the basis of measurement, and get some definite results in that particular basis, while the properties in another basis remain indefinite. But, in classical reality, all objects have definite properties in the same basis (say, positions in the same Cartesian coordinate system). This is known as the preferred basis problem, and decoherence can't explain it.
2, which is somewhat more philosophical, decoherence doesn't explain the quantitative relationship between the wave function and the probability of observing a particular outcome. Indeed, in MWI it's very hard to even define probability in a coherent way, since all outcomes actually happen and there is an uncountable (in the mathematical sense) number of outcomes.
Can you give some examples of such bases? Position is one, for sure. In "Quantum Mechanics 101" we are told that momentum is another, but I'm sceptical that there is a real experiment that can measure momentum without also measuring position.
You could perhaps create a version of QM with a quantized amplitude for the wave function in some basis.
It's also important to remember that the Schrodinger equation of a system has an infinity of solutions - for any solution, any linear function of that solution is also a solution. You need to choose a particular basis of measurement - choose a decomposition of the solutions - to be able to apply the Born rule and get from the wave function to a probability distribution.
If this is the right definition, then it's easy to disprove: for any p you chose, I can construct a series of events that each have probability q < p, but one of which is guaranteed to happen. This was the meaning of the examples I gave with the 10 thousand coins - the probability of any particular result after flipping the coins is extraordinarily small, and yet one of the results is guaranteed to happen.
Edit to add: to be clear, even if the universe itself were quantized, all measurable physical quantities, probability still wouldn't need to be quantized, as it's not a physical quantity, it's just a mathematical abstraction. Still, even in QM, not all physical quantities are quantized. For example, space (position) is not quantized in QM, and neither is time. They are both continuous quantities in all of the equations normally used. Planck time and Planck distance are only the shortest possible distances to measure precisely, given the Heisenberg uncertainty principle, but that doesn't require them to actually be quantized. In contrast, the quantization of mass, energy, spin etc are actually necessary for the theory to work, they are not just measurement artifacts.
Since all objects are composed of quantum particles, all objects are quantum objects. So a complete theory of QM needs to be able to explain, not assume, why some objects behave classically.