Does the Qur'an state that the traditions of Muhammad should be written down and followed?
Note that the Quran doesn't specify everything, for example it does say that you should pray but it doesn't give a detailed explanation of how you should pray.
Similarly the quran says that we should follow the the prophet(PBUH):
O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah and the Messenger, if you should believe in Allah and the Last Day. That is the best [way] and best in result.4:59
What are the traditions that Muhammad PB-UH have ? Muhammad had never came up with traditions but he came with values, whats a tradition ? a tradition is an attitude that one person or a group of people had developed by time regardless if it is right or wrong and their offspring had followed it without any conscious contribution, does the prophet do so ? Allah says in the Quran :
" وَنَزَّلْنَا عَلَيْكَ الْكِتَابَ تِبْيَاناً لِكُلِّ شَيْءٍ وَهُدىً وَرَحْمَةً وَبُشْرَى لِلْمُسْلِمِينَ " النحل:الآية89
this is the difference between a value and a tradition, a value is the assumption of an ethical action with regard and worth, it means after a deep conscious cooking of the idea you search just for the ethical in it and you give just the good in it, that's values as the prophet PB-UH came up with, if you search in all the Sunnah books from cover to cover you are going to find only value but never traditions.
To respond directly to your question, the Qur'an state that the values that Muhammad came up with should be written down and followed, so Allah mandated Muslims to emulate the practices of Muhammad as he practiced them. as Allah says :
قل إن كنتم تحبون الله فاتبعوني يحببكم الله ويغفر لكم ذنوبكم والله غفور رحيم (آل عمران:31)
Also to remove any other doubts about this subject we must say that any thing that Muhammad says by his tongue is a revelation, as allah says in the Quran :
وَمَا يَنطِقُ عَنِ الْهَوَى* إِنْ هُوَ إِلَّا وَحْيٌ يُوحَى" النجم : 3-4
He also says :
هُوَ الَّذِي بَعَثَ فِي الْأُمِّيِّينَ رَسُولًا مِّنْهُمْ يَتْلُو عَلَيْهِمْ آيَاتِهِ وَيُزَكِّيهِمْ وَيُعَلِّمُهُمُ الْكِتَابَ وَالْحِكْمَةَ وَإِن كَانُوا مِن قَبْلُ لَفِي ضَلَالٍ مُّبِينٍ" الجمعة : 2