I was very amused to learn that, as matters currently stand, Hillary Clinton cannot constitutionally be appointed Secretary of State. The obstacle is the Emoluments clause:
No Senator or Representative shall, during the time for which he was elected, be appointed to any civil office under the authority of the United States, which shall have been created, or the emoluments whereof shall have been increased during such time: and no person holding any office under the United States, shall be a member of either House during his continuance in office.
The compensation of the Secretary of State has been increased during Clinton’s Senate term, so she is ineligible for the office. More interestingly, there is a dispute about whether the obstacle can be removed by cutting the Secretary’s salary. (This is called the “Saxbe fix.”) A precedent dating to the Taft administration says it can, but some legal opinion disagrees and the question seems never to have been tested in court. Eugene Volokh explains.
My prediction is that Congress will institute the Saxbe fix and the Senate will confirm Clinton. Probably that will be the end of it, since it will be very difficult for opponents of the Clinton appointment (if there are any) to find someone with standing to challenge it in court.
I was very interested to read, however, that the Reagan administration believed the Saxbe fix was unconstitutional. Consequently, Reagan passed up his first choice for a Supreme Court appointment and instead picked Robert Bork. Reagan’s first choice was Orrin Hatch. Imagine how history would have been different with no Bork confirmation battle, and with Hatch on the court in place of Anthony Kennedy.
Internet Scofflaw 
You almost got it right.
There are two Emolument Clauses that I know of in the Constitution. The Emoluments Clause contained in Article I Section 9 not only states “… without the consent of Congress…”, but it also states that these prohibited appointments are “…from any King, Prince, or foreign State.” Keyword: FOREIGN.
First, Hillary (if she is chosen) will be confirmed by Congress, as are ALL Cabinet appointees, per Article II, Section 2.
Secondly, for purposes of the same cited Emoluments Clause, “a ‘foreign state’ is deemed to include an international organization in which the United States is NOT a member.” Secretery of State is not an international organization.
Source: http://micronations.wikia.com/wiki/Emoluments_clause
This Clause was unanimously added to the Constitution to prevent international conflicts of interest, and to prevent the establishment of a “special” position which would be created just to place a presidential appointee.
If you are referring to the Emoluments Clause contained in Article I Section 6, then you are, again, misinterpreting the words. It states “…and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.” Notice that “Continuance” is capitalized? The forefathers were very keen about capitilizing certain words for emphasis. Hillary can simply give up her Senate seat and will, therefore, be in compliance with the referenced Clause.
Please. Clearly I am referring to section 6 (insofar as I quoted it), and the relevant clause in this case is the one before the “and”, not the one after it. Legal experts, the historical record of the “Saxbe fix,” and the plain reading of section 6 all agree that this is an issue.