QSpace is an open-source tensor library designed to exploit quantum symmetry spaces in tensor network states for quantum many-body systems. It supports both abelian symmetries (Z_n, U(1)) and non-abelian symmetries based on semisimple classical Lie algebras (A_n, B_n, C_n, D_n), corresponding to SU(n), SO(2n+1), Sp(2n), and SO(2n). The library is implemented in C++ embedded in MATLAB via the MEX interface and is available under the Apache 2.0 license. QSpace is designed as a bottom-up approach, starting from the defining representation and Lie algebra, and explicitly computes and tabulates generalized Clebsch-Gordan coefficient tensors (CGTs) for tensor product decompositions. This allows for efficient and flexible operations across all symmetries, with symmetry-related details hidden within the C++ core libraries. QSpace automatically adapts to the needs of simulations, storing only necessary symmetry-related data. It supports a wide range of operations, including generating scalar Hamiltonian terms, handling compact symmetry labels, outer multiplicity, tensor decomposition, CGT orthonormalization, data storage, leg order conventions, and contraction of CGT networks. The library includes detailed documentation, examples, and tutorials for applications in quantum many-body simulations, such as DMRG and NRG. QSpace has evolved through several versions, with v4.0 being the latest open-source release. It provides a comprehensive framework for implementing physical models with symmetries, including handling of tensor contractions, symmetry sectors, and multiplets. The library is designed to be user-friendly, with a focus on non-abelian symmetries, and includes conventions for tensor directions, leg order, and graphical representations. QSpace is particularly useful for handling complex tensor network states with symmetries, enabling efficient numerical simulations and detailed physical insights.QSpace is an open-source tensor library designed to exploit quantum symmetry spaces in tensor network states for quantum many-body systems. It supports both abelian symmetries (Z_n, U(1)) and non-abelian symmetries based on semisimple classical Lie algebras (A_n, B_n, C_n, D_n), corresponding to SU(n), SO(2n+1), Sp(2n), and SO(2n). The library is implemented in C++ embedded in MATLAB via the MEX interface and is available under the Apache 2.0 license. QSpace is designed as a bottom-up approach, starting from the defining representation and Lie algebra, and explicitly computes and tabulates generalized Clebsch-Gordan coefficient tensors (CGTs) for tensor product decompositions. This allows for efficient and flexible operations across all symmetries, with symmetry-related details hidden within the C++ core libraries. QSpace automatically adapts to the needs of simulations, storing only necessary symmetry-related data. It supports a wide range of operations, including generating scalar Hamiltonian terms, handling compact symmetry labels, outer multiplicity, tensor decomposition, CGT orthonormalization, data storage, leg order conventions, and contraction of CGT networks. The library includes detailed documentation, examples, and tutorials for applications in quantum many-body simulations, such as DMRG and NRG. QSpace has evolved through several versions, with v4.0 being the latest open-source release. It provides a comprehensive framework for implementing physical models with symmetries, including handling of tensor contractions, symmetry sectors, and multiplets. The library is designed to be user-friendly, with a focus on non-abelian symmetries, and includes conventions for tensor directions, leg order, and graphical representations. QSpace is particularly useful for handling complex tensor network states with symmetries, enabling efficient numerical simulations and detailed physical insights.