This paper is focused on privacy issues related to the prefix part of IPv6 addresses. Long-lived prefixes may introduce additional tracking opportunities for communication partners and third parties. We outline a number of prefix alteration schemes that may be deployed to maintain the unlinkability of users' activities. While none of the schemes will solve all privacy problems on the Internet on their own, we argue that the development of practical prefix alteration techniques constitutes a worthwile avenue to pursue: They would allow Internet Service Providers to increase the attainable privacy level well above the status quo in today's IPv4 networks.