This is a really broad answer, and I'm planning on editing in more links.
- Are politicians planning on the number of cases dropping permanently and then quarantine ending?
This strategy is known as "elimination," where cases drop to a number small enough to be quarantined (with support from the government) long enough to stop transmission. This is how New Zealand succeeded. The only way for cases to drop permanently is herd immunity (which everyone is very, very far from) or preventative measures that a large conservative section of the country is rejecting. It seems most states have given up on preventative measures both in the population and in leadership, so this cannot be the goal.
On herd immunity: The basic problem with herd immunity is that we need a large number of people to get sick. In order to control an epidemic, one person has to infect fewer than one person on average. So with no preventative measures, an infected person with COVID infects three1 others, then more than 67% of those people need to be vaccinated in order to create only one new infection. If we assuming getting COVID means you're immune to reinfection, then more than 200 million people would have to get sick. And there's not room in this answer to touch on the uncertainty surrounding immunity!
Experts estimate that in the U.S., 70% of the population — more than 200 million people — would have to recover from COVID-19 to halt the epidemic. If many people become sick with COVID-19 at once, the health care system could quickly become overwhelmed. This amount of infection could also lead to serious complications and millions of deaths, especially among older people and those who have chronic conditions. [Mayo Clinic]
We may be at 20 million infections (less than 10%). At 126k deaths, we would be looking at 1,260,000 deaths to reach herd immunity. Even more once you factor in overwhelming hospitals.
- Are politicians planning on a cure being discovered and then quarantine ending?
We are not very good at "curing" viral infections, which is why we vaccinate for them so heavily. Antivirals are limited in scope, so we focus on treating the symptoms, like pneumonia. I have not seen any infectious disease expert posing this as our way out.
- Are politicians planning on a vaccine being discovered, the[n] manufactured, the[n] distributed, and then [quarantine] ending?
Basically, yes. Dr. Fauci believes we will have a vaccine by the end of the year or early 2021. In early stages of COVID, we were worried that vaccines for coronaviruses were hard or impossible to make, because we haven't made one before. However, we haven't focused this hard on one before. Obviously manufacturing and distributing is a monumental task, but efforts are underway.
Luckily, we can combine a vaccine with other preventative measures to get out of it sooner. If wearing masks reduces person-to-person transmission to two new infections, we only have to vaccinate 50%+ of the population to slow the spread significantly. By targeting highly exposed people (nurses, doctors, cashiers, etc.) we can be even more effective. With really smart vaccine distribution, it may be possible to control this epidemic with less than 100% coverage.
- Do any of these expectations include anything to do with mutations?
Compared to other viruses, the family of coronaviruses don't mutate very fast. This suggests SARS-CoV-2 is less likely to mutate fast enough to invalidate a vaccine. This also bodes well for our ability to treat it, and we don't expect death rates to spike for mutation reasons (compared to refusals to implement preventative measures that causes effective treatment to decline).
In conclusion, the plan in the US seems to be to ride it out until there is a vaccine. Because states cannot close their borders (the federal government could enforce this, if they cared to reduced deaths), it is not realistic to reduce infections in an area enough to return permanently to normal. We will have to continue with prevention measures like masks and social distancing until there is a vaccine.
Dr. Fauci was recently quoted in the New York Times, implying slowing the outbreak until a vaccine is the goal (emphasis added).
“You have an individual responsibility to yourself but you have a societal responsibility because if we want to end this outbreak, really end it and then hopefully when a vaccine comes and put as a nail in the coffin, we’ve got to realize that we are part of the process,” Dr. Fauci said, noting that some states are doing better than others.
Because we aren't sure you can only get COVID once, we may have to continue vaccinating against it like the seasonal flu (or more frequently) until each case can be stomped out as soon as it's detected.
1: The CDC currently estimates that with no measures, a person with COVID infects two to three others. I used 3 here as a round number and to compare with two later. Note that the 67% number is napkin math, it's not like a vaccination rate of 68% would magically get us out of the pandemic.