We present a model of classical binary time series derived from a matrix product state (MPS) Ansatz widely used in one-dimensional quantum systems. We discuss how this quantum Ansatz allows us to generate classical time series in a sequential manner. Our time series are built in two steps: First, a lower-level series (the driving noise or the increments) is created directly from the MPS representation, which is then integrated to create our ultimate higher-level series. The lower- and higher-level series have clear interpretations in the quantum context, and we elaborate on this correspondence with specific examples such as the spin-1/2 Ising model in a transverse field (ITF model), where spin configurations correspond to the increments of discrete-time, discrete-level stochastic processes with finite or infinite autocorrelation lengths, Gaussian or non-Gaussian limit distributions, nontrivial Hurst exponents, multifractality, asymptotic self-similarity, etc. Our time series model is a parametric model, and we investigate how flexible the model is in some synthetic and real-life calibration problems.