In the lunar daytime, no chance unless they cover many square kilometers of the lunar surface. They’d just show up as part of the apparently white surface of the Moon. And would be so small as to be invisible. Even the Hubble space telescope could only just spot the ISS on the Moon but no chance of recognizing it as it would be just over a pixel across. That’s with Hubble. So you can see that our naked eyes would have no chance at all of seeing even the most gigantic of buildings on the Moon.
The easiest way to see them would be during the local lunar night. You could see a greenhouse if it was of order one kilometer across, lit up at night. A greenhouse the size of a typical large field in the dark part of a crescent Moon would be similar in brightness to a sixth magnitude star, just visible in good conditions.
You couldn’t see an ice skating rink or a giant slalom ski slope.
The largest ski resorts are thousands of acres. If we had a fully developed ski industry on the Moon - if low gravity skiing really took off, they would only need to cover 10 acres or so to be visible as a sixth magnitude star from Earth. Or 25 acres for a fourth magnitude star, or 62.5 acres for a second magnitude star, or 157 acres for a zero magnitude star. That is, if lit to be as bright as daylight, and if it has a transparent cover, which may be a big "if".
As for street lights, the street lights of the city of Chicago on the Moon would be equivalent to a 9th magnitude star in brightness. So you'd need powerful binoculars to see them from a dark site on the Earth, and a small telescope to see them from a brighter site from the astronomical magnitude scale. That is assuming all the light is directed downwards to illuminate the streets, and that all that light is then returned back into space - in practice, it would be reduced further by the albedo of the surface. If it illuminated lunar soil, it would be reduced in brightness by an extra 2.3 magnitudes, to over 11th magnitude.
MAKING THEMSELVES VISIBLE TO US
A one megaton explosion on the Moon would be of magnitude -10 approximately. Far brighter than Sirius and approaching the brightness of the Moon itself at -12.74, equivalent to an 8% illuminated crescent Moon.
You'd see it easily. But only for a fraction of a second - you would have to be looking at the Moon at just the right moment to see it.
You can see a 1 watt blue laser from the ISS as a negative magnitude star, you can surely shine lasers from the Moon to the Earth to be visible from Earth - the main question is just now many of them you need.
See also:
Which I wrote up as: Could You See Moon City Lights Or A Greenhouse From Earth? Just For Fun
WHAT ABOUT FEATURES VISIBLE ON THE SUNLIT SIDE OF THE MOON?
If you want it to be visible against the fully illuminated Moon, then we need to look at visual acuity ( a point “star like” source won’t do).
So then, I think best bet would be to exploit the way we can see straight lines easily. As for resolution, to be easy to see with the naked eye, needs to be about 1 arc minute, the Moon is half a degree in width, so about a thirtieth of the width of the Moon so about 100 kilometers in diameter. That's for someone with near to perfect vision.
But then you'd need several of those marks to make a clear straight line, otherwise unless you knew the Moon very well you wouldn't know it is artificial .
That is unless you can make a spot so small a distinctive artificial colour which I don't know if it is possible (would our colour receptors respond to a dot of colour so small?). If you could, maybe one spot will do.
Anyway, gives a rough idea. So perhaps if you made a line of four spots each a bit over 100 kilometers in diameter and made them - maybe very bright, maybe bright and coloured, or also very dark, maybe someone with keen eyesight would see an artificial seeming feature?
This chap assumes 3 arc minutes so then ends up with the feature a bit over 300 km in diameter.
What is the smallest feature on the Moon, that can be seen from the Earth with the unaided eye?
If you can use ten times binoculars, it only needs to be 10 - 30 kilometers in diameter.
If you can use Hubble (very unlikely of course that they'd let you) then it can be 60 meters in diameter but that's for a single pixel, if you need several, you are talking about the size of the ISS, something that large might be visible to Hubble on the surface.
HubbleSite - Reference Desk - FAQs
There is one way around this though. If you can make something that casts a shadow - so tall but not otherwise large - it could maybe cast a shadow large enough to be seen. So your 30 kilometer diameter spot could instead be a 30 kilometer long high wall and then you'd see it when the sun is low casting long shadows, in optimal conditions.
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