The paper investigates the homogeneous stabilization of linear plants with uniform and logarithmic quantization of state measurements. To achieve quadratic-like stability, the homogeneous control system with quantized state measurements is transformed into a standard homogeneous one to obtain a sufficient stability condition. It is shown that a homogeneous feedback stabilization with a uniform quantizer may achieve only practical stability, while, in the case of a logarithmic quantizer, a global (finite-time, nearly fixed-time, exponential) stability is preserved, provided that the quantization density is sufficiently high. Theoretical results are demonstrated in simulations.