Moore's Law says that the number of parts on a chip doubles every 18 months. And from 1960 to about 2000, this meant that computers more or less doubled in speed every 18 months. So, i had an 8088 PC XT in 1987. My 386 in 1993 was 30 times faster. My Pentium II in 1997 was 30 times faster than that. My Athlon in 2002 was a disappointing 7 times faster. My 4 core Phenom II in 2008 was about three times faster, though there were four cores. The new machines aren't as spectacularly faster, but there are more cores. And even that seems to be slowing. The future of computers is more processors.
The good news is that software hasn't caught up. There are lots of interesting things that software could be doing that it isn't yet doing. So current computers could be lots smarter in the future. Lots of work is being done on smart phones. They're more or less the speed of that Pentium II i mentioned.
So, in 70 years? Computers will be quicker than they are now, but not a thousand times faster. There will be more computer cores or they will be much cheaper. Well, a Pentium II class computer can be had for about $30 new these days.
In 70 years, we'll still be able to access the sum of all human knowlege from a computer that fits in your pocket, and it will get used to view images of cats.