The goal of the LIGO project (as well as projects VIRGO and GEO-600) is to create terrestrial gravitational wave antennae which will detect bursts of gravitational radiation from astrophysical catastrophies. The sensitivity of the antennae to record the perturbations of the metric (the gravitational wave) has to be at such a level when the quantum behavior of macroscopic test masses (the key elements in the antennae) becomes essential. Therefore experimentalists have to invent and realize technologies which allow to reduce the decoherence of these test masses due to the heat bath and independently to create new methods of quantum measurements which will allow to beat certain limits of sensitivity (the so-called Standard Quantum Limits). These methods may be based on principles of Quantum-Non-Demolition measurements.