Why not? Don't entangled particles maintain opposite spin?

Entry and exit to/from earth would still need to be non-nuclear. Nuclear becomes ideal with deep space travel where you can accelerate slowly but constantly - an issue we would run into is the inability to communicate with the ship once reaching a certain distance. Space is largely empty but there is still interference and tech deteriorates over time. Communication gets solved once we figure out how to use quantum entanglement to communicate.

Two forms of nuclear propulsion I've heard of are literally releasing bombs on the back end and capturing that "wave" and, more realistically, directing radioactive emission off the back as thrust (extremely slow but consistent acceleration).