Pumped water batteries are geographically limited, since they can basically only be installed wherever you could build a dam anyways.

Yes, but you can be moving rocks up a hill on a train or cable car. Or concrete blocks up a crane. Geographic limitation solved.

Rocks and hills are equally geographically limited.

Cranes would work, but I think you’re ignoring the massive difference in mass between what the biggest cranes in the world can lift and even the smallest reservoir.

some people are trying this:

https://qz.com/1355672/stacking-concrete-blocks-is-a-surpris...

I believe there's a beta test facility under construction now.

Still pretty small compared to pumped storage - they give a value of 20 megawatt-hours, the Cruachan pumped storage scheme (which isn't particularly large) stores 7.1 gigawatt hours:

https://en.wikipedia.org/wiki/Cruachan_Power_Station

For comparison, the liquid air battery in this article is rated at 250mWh.

I assume you meant Mwh? :-)

Are they efficient ?

Pumped hydro hits about 80% efficiency. So if you’ve got the right geography for it, it’s one of the best choices available.

look into energy stored on energy invested

https://en.wikipedia.org/wiki/Energy_return_on_investment#ES...

according to the wiki, pumped hydroelectric storage has a ratio of 704 where as compressed air energy storage has a ratio of 792.

Those methods are actually quite similar, they're all ultimately related (limited by) tensile strength of materials. So costs and specific energy densities (per construction material unit) are similar. An important figure is cost/tensile strenght I guess.

(Compressed air: limited by container tensile strength; gravity batteries: limited by strength of cables)

I suspect even the constants involved are the same (given the materials are almost uniformly under nominal load), although I don't have time to investigate right now (a good curiosity research topic!).

They are storing liquid air at low pressure, so container tensile strength is not a limiting factor at all.

I believe these ones are not limited by tensile strength: https://interestingengineering.com/concrete-gravity-trains-m.... They are a bit limited by available geography, but far less than say pumped hydro.

Indeed! In this case it's the integrity and friction of the soil (of the hill) that's keeping the potential energy contained, and this hill soil is "free".

In my country, there are economic challenges: water isn't available everywhere, diverting water is not cheap, and it messes with water sharing agreements, which are a political issue in themselves.

Gravity batteries only need mass, it needn’t be water: https://youtu.be/gn5AM75AGvw

As I once heard from a Wyoming rancher, “Whiskey’s for drinking, water’s for fighting.” Water rights are treacherous to navigate.

It’s gonna get worse, too, with upstream countries building dams at the expense of downstream countries.