Palindrome language and Finite Automata Machines
Regarding probabilistic (and quantum) automata, you may start by digging this survey: https://arxiv.org/abs/1507.01988

For very recent results on quantum automata, you may check two recent works by Zachary Remscrim: https://eccc.weizmann.ac.il/author/659/

Palindrome is an (exclusive) stochastic language, but a probabilistic machine requires log-space to recognize it with bounded error. Two-way quantum automata recognizes palindrome language with bounded-error in super-polynomial time.