The most promising technique at this point is to use optical interferometry to resolve the surface of the planet. In this case, you will need two telescopes (li

The most promising technique at this point is to use optical interferometry to resolve the surface of the planet. In this case, you will need two telescopes (likely in space) separated by a baseline distance, "d". The two telescopes will require extremely precise synchronization in both spatial position and timing so that the light they collect will interfere at precisely the right phase, but if this can be achieved, the angular resolution of such a telescope in radians is wavelength/d. An earth sized planet has a diameter of ~13,000 km. To "resolve the planet", we would need to be able to distinguish one half of the planet from the other, meaning we need to see at a resolution of ~6000 km on the surface. Proxima Centauri is ~4 light years away, meaning that this requires an angular resolution of 1.5e-10 radians or 30 microarcseconds. That's over 30,000 times smaller than the angular size of a human hair held at arms length!) To achieve 30 microarcsecond resolution with our optical interferometer operating at 600 nm (visible light), we need the two telescopes to be 4 km apart, which practically doesn't sound unfeasible. Alternatively, you could construct a 4 km telescope, but that's far bigger than any optical telescope we have now or in the near future (the biggest telescope in the next 20 years will be 40 meter in diameter).