Quantum Markov Chains were defined by the first author [Non-commutative Markov chains. Proc. School Math. Phys. Camarino, 268-295 (1974)] as states \(\phi\) on the infinite \(C^*\)-tensor product A of countably many copies of a matrix algebra M, possessing an intrinsic statistical property (formulated in terms of generalized conditional expectations) which allows to determine their correlation functions in terms of an ''initial distribution'' (a state \(\phi_ 0\) on M) and ''transition expectations'' (completely positive maps \({\mathcal E}_ n:M\otimes M\to M).\) The paper contains several equivalent characterizations of quantum Markov chains, in terms of the density operators \(\exp [-h_{[0,n]}]\) of the restrictions of \(\phi\) to \(A_{[0,n]}=\otimes^{n}_{k=0}M_{(k)},\) or of ''Markovian cocycles''. More precisely, \(\phi\) is shown to be a Markov state if and only if \(h_{[0,n]}\) is an ''Ising potential'', i.e. \(h_{[0,n]}=\sum^{n}_{k=0}H_ k+\sum^{n}_{k=1}H_{k-1,k}, H_ k\in A_ k, H_{k-1,k}\in A_{[k-1,k]},\) such that \((*)\quad [H_{n- 1,n},[H_ n,.]^ k(H_{n,n+1})]=0\) for all k, or, equivalently, if and only if the modular automorphism groups \(\alpha^ n_ t\) of \(A_{[0,n]}\) associated with the restrictions of \(\phi\) have the inductive structure \(\alpha^ n_ t(a_{[0,n]})=\alpha_ t^{n- 1}({\mathcal U}^ n_ t*a_{[0,n]}{\mathcal U}^ n_ t),\) where \({\mathcal U}^ n_ t\in A_{[n,n+1]}\) has the cocylce property \({\mathcal U}^ n_{s+t}={\mathcal U}^ n_ t\alpha_{-t}\!^{n-1}({\mathcal U}^ n_ s).\) Condition (*) is rather restrictive: it holds for the usual quantum Ising model, but not for all the nearest neighbour potentials of interest for quantum statistical mechanics.