A four-dimensional quantum system (ququart) can be constructed by using the biphoton polarization states of frequency-nondegenerate spontaneous parametric down-conversion. We first discuss how to generate and characterize general single-ququart states with varying degrees of mixedness. We then discuss schemes to entangle multiple biphoton ququarts using linear optics and coincidence postselection.Two-dimensional quantum states and their entanglement (i.e., qubits and entangled qubits) have long been important in quantum information research. Recently, higher-dimensional quantum states and their entanglement (i.e., quDits and entangled quDits) have been found to be important for better understanding the nature of quantum entanglement and in several areas of quantum information research.In experiment, any two-dimensional degree of freedom of a photon can be utilized to implement a qubit although the polarization state is most often utilized due to the easy control and manipulation than any other two-dimensional photonic degrees of freedom. To implement photonic quDits, high-dimensionality provided by photonic orbital angular momentum, spatial paths, etc., are often utilized as the polarization state only provides a two-dimensional degree of freedom. But these multi-dimensional photonic degrees of freedoms are very difficulty to control and entangling schemes for individual quDits do not exist. (It is however possible to generate entangled pairs using the energy-time and position-momentum entanglement of the spontaneous parametric down-conversion photon pair.)Recently, a scheme to generate a pure four-dimensional quantum state (ququart), using the polarization state of an entangled photon pair of frequency-nondegenerate spontaneous parametric down-conversion, was reported [1,2]. This biphoton ququart scheme is quite promising for applications in quantum information since it is possible to access and switch among all four bases states linear optically [3].In this paper, we first discuss how to generate and characterize general biphoton ququart states with varying degrees of mixedness using ultrafast spontaneous parametric down-conversion. We then discuss entangling schemes for individually prepared multiple biphoton ququarts using linear optical elements and coincidence postselection.