We report progress on the Completely Positive Realization Problem (CPRP), which is the quantum analogue to the Positive Realization Problem (PRP), a central topic in linear systems theory. We show how this problem arises naturally in the context of Quantum Markov Chains, and show how its solution involves noncommutative extensions of the classical methods. We will illustrate how these methods bring us half way towards the solution and show that the CPRP is related to the existence of a certain "Semidefinite Representable" cone. We show how a Quantum Markov Chain reconstruction may be achieved given such cone. We will also discuss applications and open problems.