Mass, intrinsic (already moving) velocity of that mass or potential energy (pressure and temperature) of a static mass, efficiency of that mechanical-energy-by-converting-it-to-useful-continuous-torque, and ability to restore that original stored energy, right?
So, exactly what is the "stored energy" they are (1) going to release to convert to continuous torque?
(2) going to use off-peak available electric power to re-compress/restore to serve at the next cycle?
Pumped storage as at Niagara is near-ideal. A safe, very heavy stable liquid already delivered to the base of the suction of the safe, reliable, easily-controlled suction of the pumps at a continuous rate from a inexhaustible supply upriver that will be well above the minimum NPSH of the pumps. Immediately available, cheap, reliable hydro power from the plant just yards away to re-pump up the water to a stable stored elevation a few feet higher in the (very expensive! VERY controversial pond) above the pumped storage facility under near-unique conditions where that hydro power is NOT needed at night and so can be used to power the pumps to move the water, but the hydro flow IS available due to the Canadian-US early water conservation agreements.
Now, what fluid is being compressed at what efficiency of (original electric) power to stored energy many feet underground/underwater to vented energy (back at the surface) back to power?
If air, you have much less efficiency than with water pumps. Much lower final pressure unless you use positive-displacement compressor-style designs rather than fans - at great change in complexity! Now, if they assume that the compressed gas is "stored" underwater but pressurized by the height of the water column above the gas, then the continuous pressure relies on the collapsing "piston" of a tank with its bubble maintained by a flexible membrane or piston. If the underwater concrete tank is merely the pressure vessel for the stored HP gas, and the underwater location is merely a nearby storage location for holding the HP gas until vented back topside, then you can claim very little "pressure credit" for the outside water pressure holding the internal gasses at bay, since the inside air pressure varies. An externally-pressurized spherical concrete tank can resist external pressures, but they would need to balance the need for greater stored pressure against tremendously higher construction and installation and connection costs.
Now, I grant of course that the buoyancy problems can be solved by the simple application of "more concrete" on the base of the tank. Leakage could be solved and the ever-moving air pipeline from underwater to the topside power plant can be solved - as indeed it has been in the oil industry.
Nothing comes free except college-level "studies" by physicts who DON'T have to justify their assumptions and approximations to ANYBODY. Who are responsible for nothing to nobody for accuracy or reality except writing the application to get their next grant!
What has somewhat been successful is compressing air under ground in salt deposits, then re-venting that air into the burners of a combustion turbine topside to reduce the losses of using the CT compressor. Not a large savings, but enough to pay the design and construction costs.
But offshore? Just how much are they claiming the tanks will cost installed and hookeup?