Understanding the universal behaviors in quantum phases of matter lies among the long-term dreams of condensed matter physicists. However, relating universal behaviors to physical observables is challenging. Surprisingly, knowledge from other realms of physics can bring unique insights. In this thesis, we develop two useful tools: quantum entanglement and conformal field theory, which are widely used in quantum information science and string theory, respectively. We show how they reveal the previously unknown universal information of three different systems: topological liquid, random quantum states, and critical spin chains. These three systems are typical examples of the area law entanglement scaling phase, volume law entanglement scaling phase, and logarithmic law entanglement scaling phase, respectively. We start by examing the topological liquids. The universal features of topological liquids make them useful for topological quantum computation and quantum error correction. In such systems, a piece of universal information, the quantum dimension, is encoded in the constant term of von Neumann entanglement entropy. However, von Neumann entanglement entropy only captures the bipartite entanglement structure; more universal features can be encoded in the multipartite entanglement structures. Specifically, we investigate the reflected entropy -- a tripartite entanglement quantity. We unveil the reflected entropy yields a novel universal form of two-dimensional chiral topological liquid, capturing its central charge. To derive this result, we exploit the bulk-boundary correspondence, approximating the ground state of chiral topological liquid by vertex state in boundary conformal field theory. The entanglement quantities of the vertex state are then computed by explicit numerical calculation and the conformal interface method. We next study the random quantum states. The entanglement entropy of a typical state in a random ensemble exhibits the volume law scaling. Understanding the typical entanglement of random states thus provides insights into thermalization and quantum chaos. Specifically, we study von Neumann entanglement entropy of random free fermion states in the presence of ten fundamental symmetry classes. We find the fingerprint of different symmetries lies in the constant term of typical entanglement entropy and the entanglement variance. Our results establish the symmetry classification of typical quantum entanglement and the role of symmetry in quantum chaos. The discussion of boundary conformal field theory brings us to the second important tool, conformal field theory, a powerful framework for studying the critical (gapless) phases. Given a critical quantum lattice model, it is crucial yet difficult to extract the underlying conformal data. In this subject, we develop the method of wavefunction overlap, which can extract all conformal data. Our key insight is that the universal finite-size correction of the wavefunction overlap is dictated by the orbifold conformal field theory, obtained by gauging the symmetry in the replicated theory. Importantly, our method bridges the gap of operator product expansion coefficient extraction, where a numerical method was lacking. The power of our method is demonstrated by unveiling new data of the newly proposed Haagerup model built from the Haagerup fusion category. Finally, we discuss how to generalize these tools to study more exotic quantum systems, which might lead to future projects.