Our observable universe, or a subregion of our universe many times larger than the observable universe, originated from inflating from a very tiny inflationary patch. Being so small, the initial entropy has to be bounded by a very small value. But everything in our universe originated from that tiny patch after inflation, reheating, and further expansion. Does that mean almost all entropy in the observable universe is entanglement entropy if we assume the wavefunction of the universe never collapses and is linear? Because of the peculiarities of quantum mechanics, a subsystem can have far greater entropy than the system as a whole. The many worlds interpretation explains why we subjectively observe a far greater entropy, from the vantage of an Everett branch.
One also has to take account of the peculiarities of General Relativity as well when discussing the Entropy of the Universe. In fact the Entropy derives from gravitational clumping, and ultimately the formation of Black Holes with all matter fallen in. The early universe could be considered as low Entropy because of a general evenness of matter, which later became more clumped into Galaxies, etc.
The quantum aspect of this question seems to relate to whether the Universe is primarily "classical" - with lots of decoherence or (if you dont believe that "collapse") having taken place; or primarily quantum. My understanding was that most physicists considered it to be primarily classical, but there are some unresolved issues of what happened before the time of last scattering.