I'd have to think about that. They are starting to use composite materials in cranes now, but they are not widespread yet.
Strength is only one aspect. Durability is critical. If a line fails on a sailboat its an inconvenience, if it fails on a crane it's life threatening. Look at the boom hoist cable that failed in New York recently. A laborer was killed.
We looked at using composite materials to make lattice booms. Very light and strong, but we were worried about them being damaged during handling and failing at a later time. Repair is easy with steel, but not so much with carbon fiber.
Hoist ropes work at a 5 to 1 safety factor too, and if you apply that to synthetic rope it rapidly increases in diameter. That would require much larger drums. They also absorb water, so you have a potential issue with sheave (roller) width variances.
We often use synthetic slings for rigging loads. They are very light and easy to handle compared to wire, but are incredibly easy to damage either through rough handling, poor storage or simply getting cut on a sharp edge during lifting. They are also sensitive to heat and chemicals and can deteriorate when exposed to UV for long periods. That's not too much of a problem with slings, but hoist ropes are open to the elements at all times. Replacing them periodically on an expensive sail boat is one thing... On a fleet of large cranes it would be prohibitively costly.
It'll happen one day, but I think it's a long way off, especially in these monster cranes. Too much riding on what is not that big of a maintenance cost.
“The problem with having an open mind, of course, is that people will insist on coming along and putting things in it.”