  This paper consists of $3$ parts. The first part only considers classical
processes and introduces two different extensions of the notion of hidden
Markov process. In the second part, the notion of quantum hidden process is
introduced. In the third part it is proven that, by restricting various types
of quantum Markov chains to appropriate commutative sub-algebras (diagonal
sub-algebras) one recovers all the classical hidden process and, in addition,
one obtains families of processes which are not usual hidden Markov process,
but are included in the above mentioned extensions of these processes. In this
paper we only deal with processes with an at most countable state space.
