The deadline for submitting entries to the Gore-Branson carbon sequestration idea contest has passed, but, nonetheless, here is my submission. It is simple, and more importantly it is non-exotic.
The basic concept is this: While protecting existing forests even more strongly than we do now, we must greatly expand the number of acres dedicated to tree farming, and plant those acres with extremely-fast-growing pulp trees. Weyerhaeuser Corporation has genetically modified species of trees to grow very, very fast.
As we increase tree farming, we should change our economy back to one which makes MASSIVE use of paper and other cellulose-based products -- fit cellulose back into the economy whenever and wherever we can find a niche for it.
Next, instead of recycling paper, we need to BURY paper and those cellulose products after use, when it is time to throw them away -- and use the used paper and cellulose as backfill to fill exhausted strip mines.
This has the impact of sequestering carbon in the form of the relatively stable cellulose moloecule underground.
The existing paper-collection-and-recycling system can easily be converted into a a paper-collection-and-burying system. Rebuild railroad rights-of-way to exhausted strip mines, and simply dump enormous quantities of paper from the bottoms of hopper cars into the mines.
The mines can be lined (with cellulose-based sheathing materials) to cope with leachates, and they can be structured to "chimney" methane to a central collection point, and then covered and the methane recovered and converted to electrical generation and the electricity sold to the grid.
Increases in economic activity in a cellulose-burying economy would increase the rate of carbon sequestration. Paper non-recycling and strip mining become blessings, not a curse.
Fundamentally, there's nothing technologically novel about the approach. Fundamentally, it is conservative in its logic, and cheaply executed.