My information may be dated, but I don’t believe Harvard offers engineering as a major. The Harvard engineering types I know majored in applied math. Princeton has famously good engineering, and Yale has an engineering school but it is not considered nearly as highly as Stanford or MIT. These two are the powerhouses in engineering, though of course it will vary depending on what subfield of engineering you choose. There are state schools with extremely strong engineering programs (Berkeley).
The same holds true for computers specialties. MIT, Stanford are the biggies, along with CMU. Berkeley is good too. Note here that Brown has devoted a lot of resources to math and computer science, and is excellent in these areas. Yale has always been known as a humanities school, and though it is trying to catch up, it is behind HPSM. For the many who are more knowledgable about these areas than I am - please forgive me if this description is not first rate. I pick this stuff up from my kids, who are (or are about to be) math/CS majors/minors.