Why not rebuild Alcatraz to hold the world's worst criminals?

 Because we already have the Alcatraz of the Rockies, ADX Florence.

The Administrative Maximum Facility is the federal government’s supermax prison. Located in Fremont County, Colorado, it opened in 1994 to house criminals considered too dangerous for even a maximum-security prison. Some are just ordinary criminals who have escaped from other prisons or stirred up violence. Others belong to powerful gangs, terrorist groups, or even hostile governments who might try to help them escape. Notable inmates and former inmates include:

• Barry Mills and Tyler Bingham, Aryan Brotherhood
• Zacarias Moussaoui, September 11th co-conspirator, and Ramzi Yousef, architect of the 1992 World Trade Center bombing
• Dzhokhar Tsarnaev, Boston Marathon bomber
• Ted Kaczynski, the Unabomber
• Timothy McVeigh and Terry Nichols, Oklahoma City bombing
• Robert Hanssen, Russian agent inside the FBI counter-intelligence division
• Vincent Basciano, boss of the Bonanno crime family
• Richard McNair, three-time escapee who once mailed himself out of a federal prison

Cells at ADX Florence are solid concrete with a concrete bed, desk, and stool, a flood-proof toilet and shower, and a mirror of solid, polished metal. Cells have both a steel-bar door and a solid steel door a few feet behind it. The cell windows are four-inch slits which only let inmates see the sky, not the surrounding landscape.

Prisoners spend 23 hours a day in their cells, and are escorted by three guards to their hour of daily exercise in a private room, at the bottom of a swimming-pool-like pit. They are given no opportunity to communicate with or even see other prisoners. If they somehow get out of their cells and away from the guards, there’s still 24-hour video surveillance, motion detectors, and 1,400 remote-controlled steel doors throughout the facility. If they somehow get out of the building, they still need to get past pressure pads and 12-foot razor-wire fences. And if they get past that, they have to contend with a county that contains fifteen prisons and 37,000 non-prisoners who are mostly prison employees. It’s not an easy place to blend in. (Thanks to Keith Shannon for pointing this out in the comments.)

ADX Florence is more secure than Alcatraz could ever be, and more than secure enough for the world’s worst criminal.

Is it true that we wouldn't be able to rebuild the pyramids today?

 Actually, we could. But no one would be willing to pay for it.

The Great Pyramid of Giza was built around 2600 BC.
By the time Cleopatra, the last queen of Egypt, died, the pyramid was already over two and a half millennia old. It remained the world's tallest building for about 3800 years, until it was dethroned by Lincoln Cathedral in 1311, although in other respects it dwarfs the Cathedral and most modern buildings.

I say all this to give you a sense of perspective. For thousands of years, people have looked at the pyramids and wondered how they were built.

Nothing like it had been built before or since. They were filled with indecipherable hieroglyphics, a product of an ancient and unknown culture. Their constructions have even been attributed to giants, angels, or aliens.

It's all because they were so incredibly huge, and for a long time, no one had any idea how they were built.
But with the deciphering of hieroglyphics and the advancement of archaeology as a field of study, we now have a good idea of ​​how they were built.

The pyramids were great public monuments of a wealthy civilization with an abundance of surplus labor.

They cut blocks of stone and floated them down the river.

They organized hundreds and thousands of workers and moved the blockades.

They placed them in a pyramid shape, because it was the simplest shape the structure could take.

There is nothing technically challenging about pyramids, except organizing so many people toward a single goal.

These were built in honor of deities. But they were still built with simple tools and simple methods.
If you could somehow organize that many people again, there's nothing stopping you from building another pyramid.

And if you had the money, you could just use modern equipment to build them 100 times faster.