The virtual currency started as a relatively niche method of payment, devised by an anonymous programmer, but can now be used for anything from online gambling to pizza delivery.
The payment method caught the attention of mainstream investors earlier this year after it soared in value, jumping from about US$10 at the start of 2013 to a high of US$260 on April 10, buoyed by concerns over the euro's stability and retailers' increasing acceptance of Bitcoins.
This turned out to be a bubble, and Bitcoins have fallen back below the US$90 mark.
The digital currency has met considerable opposition from regulators, who are concerned it could be hijacked for money-laundering, and that it has no foundation in the real world.
Bitcoin's developers anticipated some of these concerns at the outset, and guaranteed that the amount of code that could be mined is limited. More recently, popular exchanges like Mt Gox have moved to reassure authorities that they will be subject to the same kind of regulation as more established currencies.
The Winklevoss twins hope that bringing Bitcoins on to a traditional stock exchange will help to clean up their image further, and boost liquidity by allowing anyone with a share-dealing account to buy and sell the coins.
"For many investors, the shares will represent a cost-effective and convenient means to access exposure to Bitcoins," they said in their 74-page S-1 filing lodged with the Securities and Exchange Commission.
"It eliminates the friction of buying and reduces the risks associated with storing Bitcoin while offering similar investment attributes to direct ownership," Tyler Winklevoss told the New York Times.
"The trust brings Bitcoin to Main Street and mainstream investors to Bitcoin."
This in turn could fuel another surge in Bitcoins' value, and the Winklevoss twins' own holdings. The pair said in April that they had invested as much as US$11 million in the digital currency, thought to be more than any other investor.
Each share in the Winklevoss Bitcoin Trust would be worth about a fifth of the price of a single Bitcoin and would track their weighted average price on specialist exchanges such as MT Gox KK, Bitstamp and BTC-e.
However, the S-1 also raises red flags about many of the issues which have made some sceptical about Bitcoin.
The currency could plummet at a later date if the developers changed the terms and removed the limits on how much source code could be mined, the twins said in their statement. If hackers gained enough control, they could also "alter the source code and Blockchain on which the Bitcoin Network and all Bitcoin transactions rely", the filing added.
Meanwhile, the currency still faces an existential threat from regulators, who are worried by the rapid growth of intangible digital currencies, and the potential for abuse by money launderers.
