Can we hypothetically connect the Moon and Earth via a cable?

YES, It can be possible in future and many organizations working on it. when i was searching on this same topic, and then i found presentation on linkedin slide share. must read its interesting. first we have to know about concept. 1. read more

If the cable was infinitely strong, the Earth and Moon would be drawn together in about 239 hours assuming no change in rotation (the numbers don't take axial tilt into account, don't nitpick). Whoever connected the Earth and Moon would be in a lot of trouble for about 239 hours before we all died. read more

And the free end "above" Earth would not point at Earth, it would trail by thousands of kilometers behind the Earth. We can eliminate the last two problems by extending a cable from the Lagrange points. Half of the cable would extend toward Earth, the other half away from the Earth. This would balance the mass at the Lagrange point. read more

Since the moon is not in geostationary orbit around the earth, that is: it doesn't orbit the earth at the same speed that the earth rotates (aka lunar month is different from earth day), thus the moon appears over different places of earth. Any cable attached to earth and the moon would wrap around earth over time. read more