This paper presents a method for efficiently simulating the dynamics of multipartite quantum systems with limited entanglement using a classical computer. The key idea is that the computational resources required to simulate a pure state of n qubits grow linearly with n and exponentially with the amount of entanglement. The authors show that a pure-state quantum computation can only achieve an exponential speed-up over classical computations if the entanglement increases with the size of the computation. They establish an upper bound on the maximum speed-up achievable by a quantum computation in terms of the amount of entanglement. The paper discusses the difficulty of simulating quantum systems with classical computers, as the number of complex numbers needed to describe a pure state of n qubits grows exponentially with n. However, for systems with limited entanglement, the simulation can be done efficiently. Examples of such systems include fermions with only quadratic interactions and qubits acted upon by gates from the Clifford group. The authors introduce a local decomposition of the state into tensors and vectors, which allows for efficient simulation of quantum dynamics. This decomposition is particularly useful when the entanglement is limited, as it reduces the number of parameters needed to describe the state. The simulation can be updated efficiently when single-qubit or two-qubit gates are applied, with the computational cost depending only on the amount of entanglement. The paper also discusses the implications of these results for the study of multipartite entanglement. It shows that the amount of entanglement in a quantum system directly affects the computational cost of simulating it with a classical computer. The authors propose using measures of entanglement, such as the logarithm of the Schmidt rank, to quantify the entanglement of a pure state. The results are extended to mixed states, where both quantum and classical correlations are considered. The paper concludes that efficient classical simulation of quantum computations is possible when the amount of entanglement or correlations is limited. This has implications for the study of quantum systems and the development of efficient simulation techniques for quantum phenomena.This paper presents a method for efficiently simulating the dynamics of multipartite quantum systems with limited entanglement using a classical computer. The key idea is that the computational resources required to simulate a pure state of n qubits grow linearly with n and exponentially with the amount of entanglement. The authors show that a pure-state quantum computation can only achieve an exponential speed-up over classical computations if the entanglement increases with the size of the computation. They establish an upper bound on the maximum speed-up achievable by a quantum computation in terms of the amount of entanglement. The paper discusses the difficulty of simulating quantum systems with classical computers, as the number of complex numbers needed to describe a pure state of n qubits grows exponentially with n. However, for systems with limited entanglement, the simulation can be done efficiently. Examples of such systems include fermions with only quadratic interactions and qubits acted upon by gates from the Clifford group. The authors introduce a local decomposition of the state into tensors and vectors, which allows for efficient simulation of quantum dynamics. This decomposition is particularly useful when the entanglement is limited, as it reduces the number of parameters needed to describe the state. The simulation can be updated efficiently when single-qubit or two-qubit gates are applied, with the computational cost depending only on the amount of entanglement. The paper also discusses the implications of these results for the study of multipartite entanglement. It shows that the amount of entanglement in a quantum system directly affects the computational cost of simulating it with a classical computer. The authors propose using measures of entanglement, such as the logarithm of the Schmidt rank, to quantify the entanglement of a pure state. The results are extended to mixed states, where both quantum and classical correlations are considered. The paper concludes that efficient classical simulation of quantum computations is possible when the amount of entanglement or correlations is limited. This has implications for the study of quantum systems and the development of efficient simulation techniques for quantum phenomena.