Quantum mechanics per se does not suggest reincarnation. However, quantum entanglement, Q.E., provides a speculative basis to consider. This is, of course, venturing far into wild conjecture. I apologize in advance for any conflict with a reader's religious views.
In the early years, Q.E. was considered as between two particles. Several years ago (I don't have a reference), it was discovered that three particles could participate in the same entanglement. Conceivably, this would need a fourth particle to maintain up/down spin relationships. To me, this means that any number of particles could be connected via Q.E.
Let's hypothesize that groups of people are entangled in varying degrees and brains operate on Q.E. This network of connected brains could be viewed by some as the 'mind of God', where each individual brain operates as a 'neuron' of sorts. The 'mind' could even have a memory. This is where the concept of reincarnation could fit.
Under this view, we might consider any religion as consisting of a group of believers who are strongly connected via Q.E. In this sense, all religions have a valid basis, a sense of shared thoughts. The power of group prayer could be enormous. The connected power could be huge for some individuals who become elevated in a religion.