  We investigate quantum entanglement of three photons performing discrete-time
quantum walk (DTQW) on the nearest-neighbor sites of a square lattice. Such a
DTQW setup has already been proposed to be realizable using linear optical
elements. Each lattice site corresponds to the sequential action of a half-wave
plate and a polarizing beam splitter. Two-dimensional DTQW can be mapped to an
effective one-dimensional DTQW with a four-sided coin that depends on the
polarization and direction of motion of the photons. It is known that such a
system is capable to spatially entangle two photons, prepared in a separable
state initially. We generalize the result to the case of three photons to
explore if distinct entanglement classes, namely W and GHZ states, can be
produced by this setup. In addition to spatial entanglement, we analyze the
possibility of entanglement in the polarization basis, too. Our results can be
significant for a simple and fast generation of multipartite entangled photonic
states demanded by practical quantum technology applications.
