Imagine a world in which most of the states, rather than just a few, had nuclear weapons and long-range missiles. Would it be more peaceful or war-prone?

It would not be peaceful; rather it would be war-prone. Proliferation is the spread of ABC‐weapons into the hands of more and more actors. A number of realists state that proliferation and the ensuing assurance of destruction would lessen wars (e.g. The Cold War). Others argue that through miscalculation, accident or terrorism, disasters are more likely to occur.
The Non‐Proliferation Treaty (NPT) of 1968 created a structure of controlling the increase of nuclear equipment and the like. Mutually assured destruction (MAD) refers to the state in which neither part can stop the other from destroying it. Strategic Defense Initiative (SDI) also known as “Star Wars” is a plan by the U.S. to try to establish a defense organization to bring down ballistic missiles.
Comprehensive Test Ban Treaty (CTBT), 1996, was started to halt all nuclear trial explosions. The treaty of non-proliferation treaty is a worldwide treaty whose main objective is prevention of the spread of weapons technology and nuclear weapons. The NPT also encourages teamwork during the quiet use of nuclear power and to advance the objective of achieving nuclear, all-purpose and absolute disarmament. It has a sum of 190 states with Russia, the United  Kingdom, United States, China and France being the nuclear-weapon states. (Goldstein and Pevehouse, page 217)

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The treaty is a three-pillar system, which has an implicit stability among them i.e. non-proliferation, disarmament and the freedom to quietly exercise nuclear equipment. The central bargain of NPT is for states to consent in sharing the profit of quiet nuclear expertise and follow nuclear disarmament intended to eliminate the nuclear arsenals in exchange non nuclear weapon states consent by no means to buy nuclear weapon. The pact was indefinitely extended and was exclusive of conditions during an assessment meeting in New York City on 11th May, 1995, culminating booming efforts of U.S. government which were led by Thomas Graham.
Additional actions have been adopted so as toughen the treaty making it complicated for states to obtain the ability to manufacture nuclear armaments. Critics who disagree in that the NPT cannot end the proliferation of nuclear weapons or the drive to attain them have criticized it. They convey their dissatisfaction on the inadequate nuclear disarmament progress. The treaty has actually not worked well. NPT has permitted five states, which had tested nuclear weapons to keep them for the time being though it has obligated them under article VI to reduce and in the long run reduce them.
The reduction of nuclear weapons will ensure less spread and thus a peaceful state. This reduction has not been accomplished. Three important states refused to join the NPT i.e. Pakistan India and Israel and they have produced nuclear weapons. North Korea, which had joined NPT as a non-nuclear-weapon state later, withdrew. It later tested a nuclear weapon, which was part of its nuclear weapons investigative activities conducted while it was still an NPT state-party. Negotiations have been made to convince North Korea to surrender its nuclear weapons but have not worked.
According to Kenneth Waltz article on Iran and the bomb, in the July/August 2012, U.S. and officials of Israel stated that a nuclear-armed Iran is a highly terrifying prospect, and still a threat. In actual fact, by creating a more durable balance of military power in the East, a nuclear Iran would indeed yield further stability, not a lesser amount of stability. It is very unlikely that a nuclear-armed Iran would intentionally employ a nuclear gadget. (Kenneth Waltz-July 2012 issue)
Work cited
Kenneth. “Why Iran Should Get the Bomb.” Global. N.p., July-Aug. 2012. Web. 28 Mar. 2013. 
Goldstein. “MyWebSearch Home Page.” MyWebSearch Home Page. N.p., n.d. Web. 28 Mar. 2013. 
George. “LOOKING BACK: The Nuclear Nonproliferation Treaty Then and Now.”Arms Control Association. N.p., n.d. Web. 28 Mar. 2013.