Distance over the horizon can be estimated using d = 3.57 x (h)^1/2. Distance in km, height in meters.

Width of the Taiwan Strait = 180 km (coast to coast)

Height of antenna required for direct line of sight = 2,542 m or ~8,400 ft. This would be one antenna on a tower and one on the ground.

Tallest antenna in the world is on top of the Burj Khalifa skyscraper at 2,722 ft.

If you split the difference (so each antenna just sees the other at sea level), the tower would need to be 635m tall or 2,088 ft. So if you put a Burj Khalifa tower on each coast you could make it work.

Or you can cross the relatively shorter distance between Shenzhen and hongkong

That's definitely much easier. Crossing Shenzhen to Hong Kong is literally getting off the Shenzhen subway, crossing a bridge, and entering Hong Kong immigration and MTR.

that is way, way too far for any sort of reliable point to point microwave, even with a lot of elevation above MSL on both ends.

also the chinese version of the FCC will come along with their partyvan.

when you are dealing with a government where non-licensed non-compliant ISP type things will be shut down at OSI layer 1 by armed men with carbines, you have other problems.

Legality and party vans aside, the concept might work from Shenzhen to Hong Kong.

Not sure if you were being serious, but along these lines, I stayed in a Shenzhen hotel late last year that strangely provided unfiltered internet. Of course it wasn't advertised as such, but Google worked (redirected to google.com.hk). Given the hotel was right on the border with Hong Kong, one could only speculate how this worked and which option was easier: always-on VPN or antenna :)

I've had consistent, unrestricted access to google/facebook/videos from 1989, all over China, using a tmobile sim card registered in another country.

A financial services "startup" in sz manages to run a VPN with some sort of official permission. Not sure what sort of connections made this possible...

Yes, roaming works. Preferably you use a HK sim. At least foreign companies can buy a connection to a foreign IX that doesn't go through the firewall. You can pickup HK 4G in SZ without roaming. International traffic from China sucks in general, not only because of the firewall. You can buy better international connectivity in some cases.

My AT&T roaming in China was unfiltered too.

I was in Beijing two weeks ago, with 3 phones (AT&T, Sprint and T-Mobile). I could not get Gmail on any of them. No Facebook, Twitter, Google search, Google translate, etc. I had to fall back to Yahoo Mail (and WeChat and QQ).

In Shenzhen, there was a moment in the hotel where Google worked for about 10 minutes. I was elated but then disappointed when that brief window of openness ended.

How could you tell?

I'd suggest you run some calculations of mast heights and curvature-of-Earth for various LoS distances.

Then consider possible alternatives.