Gamma-ray polarization information is valuable in many areas of contemporary physics research. An example in fundamental sector is the phenomenon of quantum entanglement, which may be investigated by analyzing relative polarizations of three gammas originating from ortho-positronium decay. In applications, an important case is the biomedical imaging with Positron Emission Tomography (PET), where it has been shown by simulated model studies, that the polarization information, which is not exploited in existing PET systems, has the potential to improve the image quality. The polarization of a gamma photon can be determined from its Compton scattering, where it produces a recoil electron and a scattered photon. For reconstruction of Compton events, one needs position and energy-sensitive detectors, usually encompassing two layers, for detection of the electron and the scattered photon, respectively. However, in many applications where detectors are highly granular and contain many channels, such as PET, a system based on two-layer readout would be costly. In this project we will construct a new, modular detector system for gamma polarization measurements, based on single-layer Compton scattering detectors. A module will contain an array of scintillators, read-out by silicon photomultipliers. Compared to two-layer detectors, the single-layer concept offers a possibility to construct more cost-efficient, compact and versatile devices. We will assemble a sixteen-module system, which will be used in two applications: first, to evaluate experimentally for the first time the feasibility of using the information about gamma-ray polarization in PET, as an important step towards the next generation of more efficient medical imaging devices, and second, to analyze the azimuthal correlations of three gammas from ortho-positronium decay in order to investigate entanglement as a fundamental concept of quantum physics.