<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0202</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2005-12-01</periodOfReport>

    <issuer>
        <issuerCik>0000825313</issuerCik>
        <issuerName>ALLIANCE CAPITAL MANAGEMENT HOLDING LP</issuerName>
        <issuerTradingSymbol>AC</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001234361</rptOwnerCik>
            <rptOwnerName>JOSEPH ROBERT H JR</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O ALLIANCE CAPITAL MANAGEMENT CORP.</rptOwnerStreet1>
            <rptOwnerStreet2>1345 AVENUE OF THE AMERICAS</rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10105</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>SVP and CFO of General Partner</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>units rep. assign. of ben. own. of lp interests</value>
                <footnoteId id="F1"/>
            </securityTitle>
            <transactionDate>
                <value>2005-12-01</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>506</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>54.70</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>40125</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value></value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Units representing assignments of beneficial ownership of limited partnership interests in Alliance Capital Management Holding L.P. (&quot;Holding Units&quot;)</footnote>
        <footnote id="F2">Total includes 18 Holding Units acquired through distribution reinvestment under an employee deferred compensation plan.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Adam Spilka, by pwr. att'y</signatureName>
        <signatureDate>2005-12-05</signatureDate>
    </ownerSignature>
</ownershipDocument>
