This paper investigates matrix product state (MPS) representations for pure multipartite quantum states. It determines the freedom in representations with and without translation symmetry, derives canonical forms, and provides efficient methods for obtaining them. The work extends results on frustration-free Hamiltonians and the generation of MPS, and discusses the use of MPS for classical simulations of quantum systems. The paper begins with an introduction and overview of MPS, followed by definitions and preliminaries. It introduces MPS and the valence bond picture, discusses finitely correlated states, frustration-free Hamiltonians, and provides examples. The canonical form of MPS is then analyzed, distinguishing between open boundary conditions (OBC) and periodic boundary conditions (PBC) with translational invariance (TI). The paper shows that for OBC, the canonical form is unique up to permutations and degeneracies in the Schmidt decomposition. For TI and PBC, it shows that the canonical form can be decomposed into superpositions of periodic states. The paper then discusses the generation of MPS, showing how they can be used to generate multipartite entangled states. It also reviews the use of MPS for classical simulations of quantum systems, highlighting their efficiency in approximating important quantum states, such as ground states of 1D local Hamiltonians. The paper concludes with a discussion of the canonical form for TI MPS, showing that it can be uniquely determined under certain conditions. It also discusses the uniqueness of the canonical form and the energy gap in the ground state. The paper provides a detailed analysis of the canonical form for TI MPS, showing that it can be decomposed into superpositions of periodic states and that the canonical form is unique under certain conditions. The paper also discusses the use of MPS for classical simulations of quantum systems, highlighting their efficiency in approximating important quantum states.This paper investigates matrix product state (MPS) representations for pure multipartite quantum states. It determines the freedom in representations with and without translation symmetry, derives canonical forms, and provides efficient methods for obtaining them. The work extends results on frustration-free Hamiltonians and the generation of MPS, and discusses the use of MPS for classical simulations of quantum systems. The paper begins with an introduction and overview of MPS, followed by definitions and preliminaries. It introduces MPS and the valence bond picture, discusses finitely correlated states, frustration-free Hamiltonians, and provides examples. The canonical form of MPS is then analyzed, distinguishing between open boundary conditions (OBC) and periodic boundary conditions (PBC) with translational invariance (TI). The paper shows that for OBC, the canonical form is unique up to permutations and degeneracies in the Schmidt decomposition. For TI and PBC, it shows that the canonical form can be decomposed into superpositions of periodic states. The paper then discusses the generation of MPS, showing how they can be used to generate multipartite entangled states. It also reviews the use of MPS for classical simulations of quantum systems, highlighting their efficiency in approximating important quantum states, such as ground states of 1D local Hamiltonians. The paper concludes with a discussion of the canonical form for TI MPS, showing that it can be uniquely determined under certain conditions. It also discusses the uniqueness of the canonical form and the energy gap in the ground state. The paper provides a detailed analysis of the canonical form for TI MPS, showing that it can be decomposed into superpositions of periodic states and that the canonical form is unique under certain conditions. The paper also discusses the use of MPS for classical simulations of quantum systems, highlighting their efficiency in approximating important quantum states.